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Distributions of zeros and masses of entire and subharmonic functions with restrictions on their growth along the strip B. N. Khabibullin Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa
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     § 1. Introduction1.1. Problem statementsA holomorphic function $f$ on the complex plane $\mathbb{C}$, that is, an entire function $f$ of finite type
$$
\begin{equation}
\operatorname{type}[\ln |f|]:=\limsup_{z\to \infty}\frac{\ln^+|f(z)|}{|z|}
\end{equation}
\tag{1.1}
$$
of order $1$ is called an entire function $f$ of exponential type [1]–[4] (here, by definition, $\ln^+x:=\max\{0,\ln x\}$). The term “entire function of finite degree” is also widely used [5], [6]. Entire functions of exponential type are most often used in various applications of the theory of entire functions, for example, in terms of such functions the dual spaces to function spaces on subsets of $\mathbb{C}$ can be realized. Our common task is as follows: given distribution of points ${\mathrm Z}$ on $\mathbb{C}$, find conditions for the existence of an entire function $f\not\equiv 0$ of exponential type that vanishes on $\mathrm Z$, counting multiplicity (written $f(\mathrm Z)=0$), under given estimates from above on the modulus $|f|$ along a fixed line. If only the relative position of this line and the distribution of points $\mathrm{Z}$ are taken into account, then the choice of a straight line is immaterial. As such a line, one often considers the real axis $\mathbb{R}\subset \mathbb{C}$ or the imaginary axis $i\mathbb{R}\subset \mathbb{C}$. In what follows, we will consider, for the most part, the imaginary axis. This choice stems from the Malliavin–Rubel theorem [7], Ch. 22 in [3], Ch. 3.2 in [4], which is the main impediment for the present study. This result is concerned with the existence of an entire function of exponential type
$$
\begin{equation}
f\not\equiv 0, \qquad f(\mathrm{Z})=0, \qquad |f(iy)|\leqslant |g(iy)|\quad \text{for all } y\in \mathbb{R},
\end{equation}
\tag{1.2}
$$
where $g\not\equiv 0$ is a given entire function of exponential type with $g(\mathrm{W})=0$ for a given distribution of points $\mathrm{W}$. A solution of this problem for distributions of points $\mathrm{Z}$ and $\mathrm{W}$ lying completely on the positive semiaxis $\mathbb{R}^+:=\{x\in \mathbb{R}\mid x\geqslant 0\}$ was given in [7] and Ch. 22 in [3] in terms of relations between point distributions $\mathrm{Z}$ and $\mathrm{W}$. In the present paper, when dealing with the problem of existence of an entire function $f$ of exponential type from (1.2), we will generally consider relations between the distributions of points $\mathrm{Z}$ on $\mathbb{C}$ and the growth rate along $i\mathbb{R}$ of a given entire function $g$ of exponential type. Even more meaningful is the subharmonic version of the problem of the existence of an entire function of exponential type
$$
\begin{equation}
f\not\equiv 0, \qquad f(\mathrm{Z})=0, \qquad \ln |f(iy)|\leqslant M(iy)\quad \text{for all } y\in \mathbb{R}\setminus E,
\end{equation}
\tag{1.3}
$$
where $M\not\equiv -\infty$ is a given subharmonic function of finite type
$$
\begin{equation}
\operatorname{type}[M]:= \limsup_{z\to \infty}\frac{M^+(z)}{|z|}\in \mathbb{R}^+, \qquad M^+(z):=\max\{0,M(z)\},
\end{equation}
\tag{1.4}
$$
for the order $1$ on $\mathbb{C}$, and $E$ is a sufficiently small exceptional set. It is also natural to consider this problem in terms of relations between the distribution of points $\mathrm{Z}$ on the one hand, and the behaviour of the function $M$ and the sizes of the exceptional set $E$, on the other hand. The principal problems considered in the present paper cover more general and more stringent subharmonic versions of the original problems (1.2) and (1.3). In these problems, for a given mass distribution $\nu$ on $\mathbb{C}$ with some restrictions on it near the imaginary axis, and, for arbitrary subharmonic function $M\not\equiv -\infty$ of finite type, necessary and sufficient conditions are given under which, for a number $b\in \mathbb{R}^+$, with the corresponding vertical open or closed strip
$$
\begin{equation}
\operatorname{str}_b:= \bigl\{z\in \mathbb{C}\bigm| |{\operatorname{Re} z}|< b\bigr\}, \qquad \overline{\operatorname{str}}_b:= \bigl\{z\in \mathbb{C}\bigm| |{\operatorname{Re} z}|\leqslant b\bigr\}
\end{equation}
\tag{1.5}
$$
of width $2b$ and with the middle line $i\mathbb{R}$, there exists a subharmonic function $U\not\equiv -\infty$ of finite type with Riesz distribution of masses $\leqslant \nu$ satisfying
$$
\begin{equation}
U(z)\equiv M(z) \quad\text{for all } z\in \operatorname{str}_b \text{ or } z\in \overline{\operatorname{str}}_b.
\end{equation}
\tag{1.6}
$$
We also consider a special choice of such a function $U=v+\ln |h|$ of finite type, with a subharmonic function $v$ with Riesz distribution of masses equal to $\nu$, and an entire function $h\not\equiv 0$ satisfying
$$
\begin{equation}
v(z)+\ln|h(z)|\leqslant M^{\bullet r}(z)\quad\text{for all } z\in \overline{\operatorname{str}}_b,
\end{equation}
\tag{1.7}
$$
where $M^{\bullet r}(z)$ are the integral average of $M$ over discs with centres $z$ and radii $r(z)$ which go to zero very quickly as $z\to \infty$, and
$$
\begin{equation}
v(z)+\ln|h(z)|\leqslant M(z)\quad\text{for all } z\in \overline{\operatorname{str}}_b\setminus E,
\end{equation}
\tag{1.8}
$$
where the exceptional set $E\subset \mathbb{C}$ is very small. Here, some restrictions on the distribution of masses $\nu$ near and on the imaginary axis are inevitable if the solutions are sought in simple geometric terms of logarithmic functions of intervals and submeasures (2.1)–(2.3) on $\mathbb{R}^+$ for $\nu$’s having origins in the logarithmic characteristics (1.16) of distributions of points on $\mathbb{R}^+$ from [7], and Ch. 22 in [3]. Here, the reader may change to § 1.3, but consult § 1.2 if required. 1.2. Some notation, definitions, conventionsWe often write singletons $\{a\}$ without curly brackets, (that is, we write $a$ for $\{a\}$). We also set $\mathbb{N}_0:=0\cup \mathbb{N}=\{0,1, \dots\}$, where $\mathbb N:=\{1,2, \dots\}$ is the set of natural numbers, $\mathbb{C}_{\infty}:=\mathbb{C}\cup \infty$, and $\overline{\mathbb{R}}:=-\infty \cup \mathbb{R}\cup +\infty$ are the extended complex plane and real axis with $-\infty:=\inf \mathbb{R}\notin \mathbb{R}$, $+\infty:=\sup \mathbb{R}\notin \mathbb{R}$, the inequalities $-\infty\leqslant x\leqslant +\infty$ for any $x\in \overline{\mathbb{R}}$, and the natural (order) topology. We set $\sup \varnothing:=-\infty$ and $\inf \varnothing:=+\infty$ for the empty set $\varnothing$. The symbol $0$ can also denote zero functions, measures, etc. For $x\in X\subset \overline{\mathbb{R}}$, its positive part is denoted by $x^+:=\sup\{0,x \}$. We also set $X^+:=\{x^+\mid x\in X\}$. If $f\colon S\to \overline{\mathbb{R}}$ is an extended scalar function, we let $f^+\colon s\underset{s \in S}{\longmapsto} (f(s))^+\in \overline{\mathbb{R}}^+$ denote its positive part; $f^-:=(-f)^+\colon S\to \overline{\mathbb{R}}^+ $ denotes its negative part. As usual, we write $f\not\equiv c$ if $f$ takes at least one value different from $c$ on its domain of definition. For an extended scalar function $m$ defined on a ray from $\mathbb{R}^+$, the order (of growth) of the function $m$ (about $+\infty$) is defined by
$$
\begin{equation}
\operatorname{ord}[m]:=\limsup_{x\to +\infty} \frac{\ln(1+m^+(x))}{\ln x}\in \overline{\mathbb{R}}^+
\end{equation}
\tag{1.9}
$$
and, for $p\in \mathbb{R}^+$, the type (of growth) of $m$ for order $p$ (about $+\infty$) is defined by
$$
\begin{equation}
\operatorname{type}_p[m]:=\limsup_{x\to +\infty} \frac{m^+(x)}{x^p}\in \overline{\mathbb{R}}^+
\end{equation}
\tag{1.10}
$$
(see [1], [5], [2], [8], § 2.1 in [9]). For an arbitrary function $u\colon \mathbb{C}\to \overline{\mathbb{R}}$ with radial growth function
$$
\begin{equation}
\mathrm{M}_u \colon r\underset{r\in \mathbb{R}^+}{\longmapsto} \sup\bigl\{u(z)\bigm| |z|=r\bigr\},
\end{equation}
\tag{1.11}
$$
$\operatorname{ord}[u]:=\operatorname{ord}[{\mathrm M}_u]$ and $\operatorname{type}_p[u]:=\operatorname{type}_p[{\mathrm M}_u]$ are, respectively, the order and the type of the function $u$ for the order $p$ (see [1], [2], [8], and Remark 2.1 in [9]). Functions $u$ of finite type $\operatorname{type}_1[u]\in \mathbb{R}^+$ for order $p=1$ are simply called functions of finite type (without mentioning or indicating the order $1$ in $\operatorname{type}[u]:=\operatorname{type}_1[u]$), as was done above for entire functions of exponential type in (1.1) and for subharmonic functions of finite type in (1.4). A distribution of masses is a positive Radon measure (see [10], Appendix A in [11], and Ch. 3 in [12]), and a distribution of charges is the difference of distributions of masses [13]. For distributions of masses or charges on $\mathbb{C}$, we will generally not indicate where their domains of definition. The action of the Laplace operator $\Delta$ on a subharmonic function $u\not\equiv -\infty$ in the domain of $\mathbb{C}$ defines, in the sense of the theory of distributions, its Riesz distribution of masses in this domain (see [12], [11], [14]).We will use $\frac{1}{2\pi}\Delta u$ (see (1.12)) and $\varDelta_u$ to denote the Riesz distribution of masses of the function $u$. In what follows, $D_z(r):=\{w \in \mathbb{C} \mid |w-z|<r\}$, $\overline{D}_z(r):=\{w \in \mathbb{C}_{\infty} \mid |w-z|\leqslant r\}$, and $\partial \overline{D}_z(r):=\overline{D}_z(r)\setminus {D}_z(r)$ are, respectively, the open and closed discs, and the circle of radius $r\in \overline{\mathbb{R}}^+$ with centre $z\in \mathbb{C}$. Next, $\mathbb{D}:=D_0(1)$ and $\overline{\mathbb{D}}:=\overline D_0(1)$, and $\partial \overline{\mathbb{D}}:=\partial \overline{D}_0(1)$ are, respectively, the open and closed unit discs, and the unit circle in $\mathbb{C}$. By $\mathbb{C}_{\mathrm{rh}}:=\{z\in \mathbb{C} \mid \operatorname{Re} z>0\}$, $\mathbb{C}_{\overline{\mathrm{rh}}}:= \mathbb{C}_{\mathrm{rh}}\cup i\mathbb{R}$, $\mathbb{C}_{\mathrm{lh}}:=-\mathbb{C}_{\mathrm{rh}}$, and $\mathbb{C}_{\overline{\mathrm{lh}}}:=-\mathbb{C}_{\overline{\mathrm{rh}}}$ we denote the open and closed right half-planes, and the open and closed left half-planes in $\mathbb{C}$. If $\nu$ is a distribution of charges on $S\subset \mathbb{C}$, we let $\nu^+:=\sup\{\nu,0\}$, $\nu^-:=(-\nu)^+$ and $|\nu|:=\nu^++\nu^-$ denote, respectively the upper, lower, and total variation of this distribution of charges; its support is denoted by $\operatorname{supp} \nu=\operatorname{supp} |\nu|$. The distribution of charges $\nu$ is concentrated on a $\nu$-measurable subset $S_0\subset S$ if the total variation $|\nu|$ of the complement $S\setminus S_0$ of the set $S$ is zero. The restriction of a function $f$ to $S\subset \mathbb{C}$ is denoted as $f\lfloor_S$. In what follows, $\nu\lfloor_S$ also denotes the restriction of a positive Borel measure or a charge $\nu$ to an $\nu$-measurable subset $S\subset \mathbb{C}$. For $r\in \overline{\mathbb{R}}^+$ and such $\nu$, by
$$
\begin{equation}
\nu_z^{\mathrm{rad}} (r):=\nu \bigl(\overline D_z(r)\bigr),\qquad \nu^{\mathrm{rad}}(r):=\nu_0^{\mathrm{rad}}(r)=\nu(r\overline{\mathbb{D}})
\end{equation}
\tag{1.13}
$$
we denote the radial right continuous counting functions of the charge distribution $\nu$ with centres, respectively, at $z\in \mathbb{C}$ and at zero. The upper density of a distribution of charges $\nu$ for order $p\in \mathbb{R}^+$ is equal to
$$
\begin{equation}
\operatorname{type}_p[\nu]:=\operatorname{type}_p[|\nu|] \stackrel{(1.10)}{:=} \limsup_{0<r\to +\infty} \frac{|\nu|(r\overline{\mathbb{D}})}{r^p} \stackrel{(1.13)}{=} \limsup_{0<r\to +\infty} \frac{|\nu|^{\mathrm{rad}}(r)}{r^p}\in \overline{\mathbb{R}}^+,
\end{equation}
\tag{1.14}
$$
and for $p = 1$, we omit mention of the order. In particular, a distribution of charges $\nu$ has finite upper density if $\operatorname{type}[\nu]:= \operatorname{type}_1[\nu] < +\infty$. The order of distribution of charges $\nu$ is defined by $\operatorname{ord}[\nu]\stackrel{(1.9)}{:=}\operatorname{ord} [|\nu|^{\mathrm{rad}}]$. In what follows, for distributions of (generally repeating) points on $\mathbb{C}$, it is assumed that that each disc $r\mathbb{D}$ for $r\in \mathbb{R}^+$ contains a finite number of points from this distribution of points, that is, we consider only locally finite distributions of points in $\mathbb{C}$. A distribution of charges is called integer-valued if it assumes only integer values from $\mathbb{Z}:=\mathbb{N}_0\cup (-\mathbb{N})$ on bounded sets. The distribution of points $\mathrm{Z}$ can be looked upon as an integer-valued distribution of masses, for which the mass of each $\mathbb{C}$-bounded set is equal to the number of its points that fall in $\mathrm{Z}$. For this integer-valued distribution of masses, we keep the same notation $\mathrm{Z}$. In what follows, $\mathrm{Z}$ and $\mathrm{W}$ are distributions of points on $\mathbb{C}$. Thus, each integer-valued distribution of masses uniquely defines a‘locally finite distribution of points, and vice versa, the equality $\mathrm{Z}=\mathrm{W}$ is understood as the equality of the corresponding distributions of masses (see [4], § 0.1.2). All concepts and notation introduced in this paper for distributions of charges and masses are also transferred to distributions of points. Thus, $\mathrm{Z}\lfloor_{S}$ is the restriction of the distribution of points $\mathrm{Z}$ to $S\subset \mathbb{C}$. If $\operatorname{supp} \mathrm{Z}\subset S$, then, for brevity, we will often simply write $\mathrm{Z}\subset S$. For an entire function $f\not\equiv 0$, we denote by $\operatorname{Zero}_f$ its distribution of zeros, which is a distribution of points in which each point $z\in \mathbb{C}$ is repeated as many times as the multiplicity of zero of function $f$ at point $z$. In this case, $\operatorname{Zero}_f=\frac{1}{2\pi}\Delta \ln |f|$ is an integer-valued Riesz distribution of masses of the subharmonic function $\ln|f|$ (see Theorem 3.7.8 in [11]). The entire function $f\not\equiv 0$ vanishes on $\mathrm{Z}$, and we write $f(\mathrm{Z})=0$ if the inequality $\mathrm{Z}\leqslant \operatorname{Zero}_f$ is fulfilled for integer-valued distributions of masses $\mathrm{Z}$ and $\operatorname{Zero}_f$. The distribution of points $\mathrm{Z}$, when labelled by elements from $N\subset \mathbb{Z}$, can be considered as a sequence $\mathrm{Z}=(\mathrm{z}_n)_{n\in N}$ of complex numbers, where each number $z_n=z\in \mathbb{C}$ occurs exactly $\mathrm{Z}(z)$ times, that is, the same number of times the point $z\in \mathbb{C}$ is repeated in the distribution of points $\mathrm{Z}$. We also have
$$
\begin{equation}
\sum_{\substack{z\in S\\ z\in \mathrm{Z}}}f(z):= \int_Sf\,d \mathrm{Z}=\sum_{\mathrm{z}_n\in S} f(\mathrm{z}_n)
\end{equation}
\tag{1.15}
$$
when for the numbering $\mathrm{Z}=(\mathrm{z}_n)_{n\in N}$ for $S\subset \mathbb{C}$, if, for the extended scalar function $f$ on $\operatorname{supp} \mathrm{Z}$, the integral and the sum on the right in (1.15) are correctly defined. If, for some numbering $\mathrm{Z}=(\mathrm{z}_n)_{n\in N}$ on $\mathbb{C}$, we can select $c\in \mathbb{R}^+$ and a sequence of pairwise distinct integers $(\mathrm{m}_n)_{n\in N}$ such that
$$
\begin{equation*}
\sum_{k\in N}\biggl|\frac{1}{\mathrm{z}_n}- \frac{c}{i\mathrm{m}_n}\biggr|<+\infty,
\end{equation*}
\notag
$$
then the external Redheffer density of $\mathrm{Z}$ along $i\mathbb{R}$ does not exceed $c$ (see [15]–[17], § 2.1.1 in [4], [18]), and this density is equal to the infimum of such $c\in \mathbb{R}^+$. 1.3. Available results for entire functionsThe problem of the existence of an entire function $f$ of exponential type with properties (1.2) for any entire function $g\not\equiv 0$ of exponential type that vanishes on a given distribution of points $\mathrm{W}$ was completely solved, in terms of the relations between $\mathrm{Z}$ and $\mathrm{W}$, in the early 1960s in the joint work of P. Malliavin and L. A. Rubel (see Theorem 4.1 in [7]), but only for pairs of distributions of positive points $\mathrm{Z}$ and $\mathrm{W}$ on $\mathbb{R}^+$. This solution, without any significant modifications, is presented in one of the main chapters of the 1996 book by Rubel with Colliander (see Ch. 22 in [3]). Theorem (Malliavin–Rubel). Let $\mathrm{Z}\subset \mathbb{R}^+\setminus 0$ and $\mathrm{W}\subset \mathbb{R}^+\setminus 0$ be positive distributions of points of finite upper density. Them the following three assertions are equivalent. I. For any entire function $g\not\equiv 0$ of exponential type with $g(\mathrm{W})=0$, there exists an entire function $f$ of exponential type with properties (1.2). II. There is $C\in \mathbb{R}$ such that, in notation (1.15),
$$
\begin{equation}
\sum_{\substack{r<z\leqslant R\\z\in \mathrm{Z}}}\frac{1}{z}\leqslant \sum_{\substack{r<w\leqslant R\\w\in \mathrm{W}}}\frac{1}{w}+ C\quad \text{for all } 0<r<R<+\infty.
\end{equation}
\tag{1.16}
$$
III. There exist an entire function $g$ of exponential type with distribution of zeros $\operatorname{Zero}_g$ and restriction $\operatorname{Zero}_g\lfloor_{\mathbb{C}_{\mathrm{rh}}}=\mathrm{W}$ and an entire function $f$ of exponential type such that (1.2) holds. Similar results were obtained by the author of the present paper in 1988–1989 for complex $\mathrm{Z} $ and $\mathrm{W}$ on $\mathbb{C}$ in a form close to (1.2), but with some small addition to the right-hand sides in (1.2). So, for $g\equiv 1$, the following result holds. Theorem 1 (see [19], the main theorem). For a distribution of points $\mathrm{Z}\subset \mathbb{C}$ on $\mathbb{C}$, for any $\varepsilon \in \mathbb{R}^+\setminus 0$, there exists an entire function $f\not\equiv 0$ of exponential type such that $f(\mathrm{Z})=0$ and $\ln |f(iy)|\leqslant \varepsilon |y|$ for all $y\in \mathbb{R}$ if and only if $\mathrm{Z}$ is of finite upper density and, for any $\varepsilon \in \mathbb{R}^+\setminus 0$, there exists $C_{\varepsilon}\in \mathbb{R}$ such that The proof of both Theorem 1 (see [19], the main theorem) and the Malliavin–Rubel theorem depends essentially on the special case $\mathrm{Z}\subset i\mathbb{R}$ from the 1972 paper by Krasichkov-Ternovskiĭ (see Theorems 8.3, 8.5 in [20]) on spectral synthesis. This special case $\mathrm{Z}\subset i\mathbb{R}$ was substantially strengthened by the author of the present paper (see the main theorem in [21]) with a reciprocal balance on the growth rate of an entire function of exponential type along $\mathbb{R}$ and $i\mathbb{R}$. A particular case of this result is given in the following theorem (see Theorem 1 in [21] and Theorem 3.3.8 in [4]). Theorem 2. Let $\mathrm{Z}\subset \mathbb{C}$ be a distribution of points of finite upper density such that $\operatorname{type}[\mathrm{Z}]\stackrel{(1.14)}{<}d\in \mathbb{R}^+$ and $\lim_{\mathrm{Z}\ni z\to\infty}|{\operatorname{Re} z}|/|z|=0$. Then, for any $\varepsilon \in (0,1)$, there exists an entire function of exponential type $f\not\equiv 0$ such that $f(\mathrm{Z})=0$ and For the case of a majorant $\ln |g(iy)|+ \varepsilon |y|$, with an arbitrarily small number $\varepsilon >0$ as before, and an entire function $g\neq 0$ of exponential type, the following result was established in 1989. Theorem 3 (see [22], the main theorem, [23], the main theorem, and [4], Theorem 3.2.1). Let $\mathrm{Z}\subset \mathbb{C}$ and $\mathrm{W}\subset \mathbb{C}_{\mathrm{rh}}$ be distributions of points of finite upper density. Then the following three assertions are equivalent. I. For each entire function $g\not\equiv 0$ of exponential type that vanishes on $\mathrm{W}$, and any $\varepsilon \in \mathbb{R}^+\setminus 0$, there exist an entire function $f\not\equiv 0$ of exponential type that vanishes on $\mathrm{Z}$, and a Borel set $E\subset \mathbb{R}$ of finite linear Lebesgue measure $\mathfrak{m}_1(E)<+\infty$ such that
$$
\begin{equation}
\ln|f(iy)|\leqslant \ln|g(iy)|+\varepsilon |y|\quad\textit{for all } y\in \mathbb{R}\setminus E.
\end{equation}
\tag{1.19}
$$
II. For the logarithmic submeasure of the point distribution $\mathrm{Z}$ defined by
$$
\begin{equation}
\ell_{\mathrm{Z}}(r,R)\stackrel{(1.15)}{:=} \max \Biggl\{ \sum_{\substack{r<|z|\leqslant R\\z\in \mathrm{Z}}} \operatorname{Re}^+ \frac{1}{z},\sum_{\substack{r<|z|\leqslant R\\z\in \mathrm{Z}}}\operatorname{Re}^- \frac{1}{z}\Biggr\},
\end{equation}
\tag{1.20}
$$
for any $\varepsilon \in \mathbb{R}^+\setminus 0$, there exists $C_{\varepsilon}\in \mathbb{R}$ such that
$$
\begin{equation}
\ell_{\mathrm{Z}}(r,R)\leqslant \ell_{\mathrm{W}}(r,R)+ \varepsilon\ln\frac{R}{r}+C_{\varepsilon} \quad\textit{for all } 0< r<R<+\infty.
\end{equation}
\tag{1.21}
$$
III. There is an entire function $g\not\equiv 0$ of exponential type with $\operatorname{Zero}_g\lfloor_{\mathbb{C}_{\mathrm{rh}}}=\mathrm{W}$, for which, for any $\varepsilon \in \mathbb{R}^+\setminus 0$, there exits an entire function $f\not\equiv 0$ of exponential type, $f(\mathrm{Z})=0$, and a Borel subset $E\subset \mathbb{R}$ such that $\mathfrak{m}_1(E)<+\infty$ and (1.19) holds. A passage to the more stringent, as compared to Theorems 1–3, requirement $\varepsilon =0$ naturally calls for the need, both in problems of the form (1.2) and of the form (1.3), to distinguish between the cases of convergence or divergence of the logarithmic integrals (see [24]–[26], § 1.3 in [27], and § 4.6, formula (4.16) in [28])
$$
\begin{equation}
\int_{-\infty}^{+\infty}\frac{\ln^+ |g(iy)|}{1+y^2}\,dy, \qquad \int_{-\infty}^{+\infty}\frac{M^+ (iy)}{1+y^2}\,dy.
\end{equation}
\tag{1.22}
$$
If the first integral in (1.22) is finite, then this entire function $g$ of exponential type is called a Cartwright class function along $i\mathbb{R}$. In this case, problem (1.2) can be easily covered by the Beurling–Malliavin theorems of the 1960s on multiplier and the radius of completeness (see Ch. 2 in [4], [14]–[16], and [28]–[33]) even with restrictions on the values of types of entire functions. The following result is a combination of the Beurling–Malliavin theorems motivated by the Malliavin–Rubel theorem. Theorem 4. For any $c\in \mathbb{R}^+\setminus 0$, for each distribution of points $\mathrm{Z}\subset \mathbb{C}$, the following four assertions are equivalent. I. For every entire function $g\not\equiv 0$ of exponential type and each $b\in \mathbb{R}^+$, there exists an entire function $f\not\equiv 0$ of exponential type
$$
\begin{equation}
\operatorname{type}[\ln|f|]< \pi c+\operatorname{type}[\ln|g|],
\end{equation}
\tag{1.23}
$$
vanishing on $\mathrm{Z}$ such that $|f(z)|\leqslant |g(z)|$ for all $z\in \overline{\operatorname{str}}_{b}$. II. The external Redheffer density of $\mathrm{Z}$ along $i\mathbb{R}$ is smaller than $c$. III. There exists an entire function $f\not\equiv 0$ of exponential type $\operatorname{type}[\ln|f|]<\pi c$ which is bounded on the imaginary axis $i\mathbb{R}$ and vanishes on $\mathrm{Z}$. IV. There exists a Cartwright class entire function $f\not\equiv 0$ along $i\mathbb{R}$ of type $\operatorname{type}[\ln|f|]<\pi c$ that vanishes on $\mathrm{Z}$. A simple derivation of Theorem 4 is given at the end of § 2.2. A much more subtle problem of the existence or a construction of an entire function $f\not\equiv 0$ of exponential type which vanishes on $\mathrm{Z}$ and satisfies the non-strict inequalities $\operatorname{type}[\ln|f|]\leqslant \pi c+ \operatorname{type}[\ln|g|]$ (see (1.23)) and $\operatorname{type}[\ln|f|]\leqslant \pi c$ (see assertion III of Theorem 4) was studied by the author of the present paper in collaboration with T. Yu. Baiguskarov, G. R. Talipova, and F. B. Khabibullin (see [34] and [27]) in 2014–2016. They even considered version of (1.3) for subharmonic functions $M\not\equiv -\infty$ of finite type under the condition that the second integral in (1.22) is finite, and the problem itself was solved at the criterion level. By the early 1990s, the most strong results for the version of (1.2) were obtained for the case of an entire function $g\not\equiv 0$ of exponential type with, generally, divergent first logarithmic integral in (1.22), provided that both $\mathrm{Z}\subset \mathbb{C}$ and $\operatorname{Zero}_g \subset \mathbb{C}$ are located outside of some pair of, respectively, open or closed vertical angles
$$
\begin{equation}
\mathrm{X}_a:=\bigl\{z\in \mathbb{C} \bigm| |{\operatorname{Re} z}|< a|z|\bigr\},\quad \overline{\mathrm{X}}_a:=\bigl\{z\in \mathbb{C} \bigm| |{\operatorname{Re} z}| \leqslant a|z|\bigr\}, \qquad a\in [0,1],
\end{equation}
\tag{1.24}
$$
with the bisector $i\mathbb{R}$, where ${\mathrm X}_0=\varnothing$, $\overline{\mathrm{X}}_0=i\mathbb{R}$, ${\mathrm X}_1=\mathbb{C}\setminus \mathbb{R}$, $\overline{\mathrm{X}}_1=\mathbb{C}$, the opening $2\arcsin a$ of angles (1.24) can be arbitrarily small as $0<a\to 0$. Theorem 5 (see [35], the main theorem). Let $\mathrm{Z}$ be a distribution of points on $\mathbb{C}$ and $g\not\equiv 0$ be an entire function of exponential type such that there exists $a\in (0,1)$ for which $\mathrm{Z}\subset \mathbb{C}\setminus \overline{\mathrm{X}}_a$ and $\operatorname{Zero}_g\subset \mathbb{C}\setminus \overline{\mathrm{X}}_a$. Then a necessary and sufficient condition that there exist an entire function $f$ of exponential type with properties (1.2) is that there exist $C\in \mathbb{R}$ such that, in notation (1.20), The latest advances of 2020–2022 obtained by the author of the present paper with A. E. Salimova on the problem in the form (1.2) were put forward solely within the framework of entire functions of exponential type (see [36]–[38], [18]). Some results of these studies (with appropriate modifications) are essentially used below. We do not present most of the main results from these works, since they can be obtained (after some tailoring) as special cases of the results of the present paper (see the remarks below). Several main results of this paper were reported at the International Conference on Complex Analysis Dedicated to the Memory of Andrei Gonchar and Anatoliy Vitushkin (October, 2021); the video of the report is available (see [39]). The author would like to express his deep gratitude to the referee for correcting a number of inaccuracies in the original version of the manuscript and for useful comments and advice. 1.4. Special versions of main resultsBelow, in the formulation of the two special cases of the results from § 2 and § 11, priority will be given to simplicity and brevity. These results are formulated only for entire functions of exponential type without the involvement of subharmonic functions. But even these lighter options are capable to adequately illustrate the significant strengthening of the previous results even within the framework of problem (1.2), without touching the subharmonic versions of (1.3) and (1.6)–(1.8). The next result, which is a development of Theorem 5, is close in the form to the Malliavin–Rubel theorem and Theorem 3. Theorem 6. Let $g\not\equiv 0$ be an entire function of exponential type, and let, for the distribution of points $\mathrm{Z}$, there exist a number $a\in (0,1)$ such that the external Redheffer density of the restriction $\mathrm{Z}\lfloor_{\overline {\mathrm X}_a}$ along $i\mathbb{R}$ is finite. Then the following three assertions are equivalent. I. For each $b\in \mathbb{R}^+$, there exists an entire function $f\not\equiv 0$ of exponential type such that $f(\mathrm{Z})=0$ and $|f(z)|\leqslant |g(z)|$ for all $z\in \overline{\operatorname{str}}_b$. II. If $n\in \mathbb{N}_0$ and $N\in \mathbb{N}$, then
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n<N} \biggl(\ell_\mathrm{Z}(2^n,2^N)-\frac{1}{2\pi}\int_{2^n}^{2^N} \frac{\ln |g(iy)g(-iy)|}{y^2}\,d y\biggr)<+\infty.
\end{equation}
\tag{1.26}
$$
III. There exist $p\in [0,1)$, an entire function $f\not\equiv 0$ of exponential type, $f(\mathrm{Z})=0$, an $\mathfrak{m}_1$-measurable subset $E\subset \mathbb{R}^+$ and a function $r\underset{r\in \mathbb{R}^+}{\longmapsto} \mathfrak{m}_1(E\cap [0,r])$ of order $<1$ such that The following result is a direct generalization of the Mallivin–Rubel theorem. Theorem 7. Let $\mathrm{Z}\subset \mathbb{C}$ be the distribution of points as in Theorem 6, let $\mathrm{W}\subset \mathbb{C}$ be a distribution of points of finite upper density, and let I. For each entire function $g\not\equiv 0$ of exponential type with $g(\mathrm{W})=0$ and for any $b\in \mathbb{R}^+$, there exists an entire function $f\not\equiv 0$ of exponential type which vanishes on $\mathrm{Z}$ and is such that $|f(z)|\leqslant |g(z)|$ for all $z\in \overline{\operatorname{str}}_b$. II. If $n\in \mathbb{N}_0$ and $N\in \mathbb{N}$, then
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_\mathrm{Z}(2^n,2^N)- \ell_\mathrm{W}(2^n,2^N)\bigr)<+\infty,
\end{equation}
\tag{1.29}
$$
III. For some $p\in [0,1)$, there exist an entire function $g\not\equiv 0$ of exponential type such that $\operatorname{Zero}_g \lfloor_{\mathbb{C}_{\mathrm{rh}}} = \mathrm{W}\lfloor_{\mathbb{C}_{\mathrm{rh}}}$, an entire function $f\not\equiv 0$ of exponential type with $f(\mathrm{Z})=0$, and a set $E\subset \mathbb{R}^+$, as in assertion III of Theorem 6, such that (1.27) holds. Theorem 6, which can be derived from the main results at the end of § 2.2, Theorem 7, which is a special case of Corollary 3 from § 11, and, a fortiori, the main results of the paper from § 2 and § 11 strengthen, generalize, and supplement the previously available results in several different directions. First, earlier, in all previous results, including the last ones ([37], the main theorem, [38], Theorems 2.1 and 4.3, [18], Theorems 2, A and B, Remarks 1 and 2) requirements of the form (1.2) were supplemented with various quite stringent constraints on the distribution of zeros $\operatorname{Zero}_g$ in a pair of angles $\overline{\mathrm X}_a\stackrel{(1.24)}{\supset} i\mathbb{R}$ for some $a>0$. In our Theorem 6, the entire function $g\not\equiv 0$ of exponential type is arbitrary, and, in Theorem 7, only the single condition (1.28) is imposed on the distribution of points $\mathrm{W}$, to the effect that the number of points in the right half-plane $\mathbb{C}_{\mathrm{rh}}$ is not smaller, in a sense, than that in $\mathrm{W}$ in the left half-plane. Second, restrictions on the growth of the function $|f|\leqslant |g|$ were previously considered only on $i\mathbb{R}\stackrel{(1.5)}{=}\overline{\operatorname{str}}_0$, but, in Theorems 6 and 7, these restrictions are considered on the strip $\overline{\operatorname{str}}_b$ of arbitrarily large width $2b\geqslant 0$. Third, the restrictions on the distribution of points $\mathrm{Z}$ in (1.17), (1.20), (1.21), (1.25) require a verification of the inequalities on the continuum set of all half-open intervals $(r,R]\subset \mathbb{R}^+$ with $r\geqslant 1$. But assertion II of Theorem 6 and assertion II of Theorem 7 call for a verification of the single condition (1.26) and only over the countable set of half-open intervals $(2^n,2^N]\subset \mathbb{R}^+$, $0\leqslant n<N\in \mathbb{N}$. Finally, the relaxation of the assumptions in the implication of III $\Rightarrow$ I of the Malliavin–Rubel theorem, in the implication III $\Rightarrow$ I of Theorem 3, and in the sufficient condition (1.25) of Theorem 5, with preservation of the corresponding equivalences, also seriously supplements these theorems. It is assertion III of Theorem 6 and assertion III of Theorem 7 which relax the initial assumption in three directions, where the inequalities $|f(iy)|\leqslant |g(iy)|$ for all $y\in \mathbb{R}$ are replaced by much weaker log-transformed inequalities (1.27) with integral averages over expanding circles of $\ln |g|$ with an additive increasing additive on the right outside a quite massive exceptional set $E$. Much weaker versions of these results will be presented in § 2 in the subharmonic version in assertions V of our main theorem, in assertion IV of Theorem 8, and in § 11 in assertion V of Theorem 10 and assertion IV of Theorem 11. The present paper does not consider applications of our main results to the problems of non-triviality of weighted spaces of entire functions of exponential type, completeness of exponential systems in function spaces, uniqueness theorems for entire functions of exponential type, the existence of multiplicators which suppress the growth of an entire function along a line, representations of meromorphic functions as a ratio of entire functions of exponential type with restrictions on the growth rate of these functions along a straight line, the analytical continuation of series, problems of spectral analysis (synthesis) in spaces of holomorphic functions, etc., as was done or illustrated in [4], [7], [18], [19], [21]–[23], [35], [37], [40]. The author plans to expound these applications in a separate paper. § 2. Main results2.1. Formulation of the main theoremFor a distribution of charges $\nu$,
$$
\begin{equation}
\ell_{\nu}^{\operatorname{rh}}(r, R) :=\int_{r<|z|\leqslant R} \operatorname{Re}^+ \frac{1}{z} \,d \nu(z)\in \mathbb{R}, \qquad 0< r < R < +\infty,
\end{equation}
\tag{2.1}
$$
$$
\begin{equation}
\ell_{\nu}^{\operatorname{lh}}(r, R) :=\int_{r<|z|\leqslant R}\operatorname{Re}^- \frac{1}{z} \,d \nu(z)\in \mathbb{R}, \qquad 0< r < R < +\infty,
\end{equation}
\tag{2.2}
$$
are, respectively, the right and left logarithmic functions of the intervals $(r,R]$ on $\mathbb{R}^+$. If $\mu$ is a distribution of masses, then these functions generate the right logarithmic measure $\ell_{\mu}^{\operatorname{rh}}$ and the left logarithmic measures $\ell_{\mu}^{\operatorname{lh}}$ on $\overline{\mathbb{R}}^+\setminus 0$, the value $R=+\infty$ in (2.1)–(2.2) is allowed with possible values $+\infty$ for $\ell_{\mu}^{\operatorname{rh}}(r, +\infty)$ and $\ell_{\mu}^{\operatorname{lh}}(r, +\infty)$, as well as for its two-sided logarithmic submeasure on $\mathbb{R}^+\setminus 0$ defined by
$$
\begin{equation}
\ell_{\mu}(r, R):=\max \bigl\{ \ell_{\mu}^{\operatorname{lh}}(r, R), \ell_{\mu}^{\operatorname{rh}}(r,R)\bigr\}\in \overline{\mathbb{R}}^+, \qquad 0< r < R \leqslant +\infty.
\end{equation}
\tag{2.3}
$$
If $\mathrm{Z}$ are distributions of points, then this gives exactly $\ell_\mathrm{Z}(r, R)$ from (1.20). For $d\in \mathbb{R}^+$, a function $r\colon \mathbb{C}\to \overline{\mathbb{R}}^+\setminus 0$, and the gamma function $\Gamma$, the outer measure
$$
\begin{equation}
\mathfrak{m}_d^r\colon S\underset{S\subset \mathbb{C}}{\longmapsto} \inf \biggl\{\sum_k \dfrac{\pi^{d/2}}{\Gamma (1+d/2)} r_k^d\biggm| S\subset \bigcup_k\overline D_{z_k}(r_k), \, z_k\in \mathbb{C}, \, r_k \leqslant r(z_k)\biggr\},
\end{equation}
\tag{2.4}
$$
is called the $d$-dimensional Hausdorff content of variable radius $r$ (see § II in [41], [42], § 2.10 in [43], Ch. 5 in [44], Ch. 2 in [10], [45], § 5.2 in [46], and Definition 3 in [47]). If the function $r>0$ is constant, then
$$
\begin{equation}
\mathfrak{m}_d\colon S\underset{S\subset \mathbb{C}}{\longmapsto} \lim_{0<r\to 0} \mathfrak{m}_d^r(S) \underset{r>0}{\geqslant} \mathfrak{m}_d^r(S) \geqslant \mathfrak{m}_d^\infty(S),
\end{equation}
\tag{2.5}
$$
is the $d$-dimensional Hausdorff measure, which is a regular Borel measure. From definition (2.4) it is clear that
$$
\begin{equation}
\mathfrak{m}_d\geqslant \mathfrak{m}_d^{r}\geqslant \mathfrak{m}_d^{t}\geqslant \mathfrak{m}_d^{\infty} \quad\text{for any pairs of functions } r\leqslant t.
\end{equation}
\tag{2.6}
$$
In particular, $\mathfrak{m}_2$ is the planar Lebesgue measure on $\mathbb{C}$, and, for any Lipschitz curve $L$ in $\mathbb{C}$, the restriction of $\mathfrak{m}_1\lfloor_L$ is a measure of the arc length on $L$ (see § 3.3.4A in [10], which is quite consistent with the previous notation $\mathfrak{m}_1$ for the linear Lebesgue measure on $\mathbb{R}$). Note that $\mathfrak{m}_d=\mathfrak{m}_d^r=0$ for each $d>2$. In addition, the $0$-dimensional Hausdorff measure $\mathfrak{m}_0$ of a set is the cardinality of this set, and, for all $d>2$, both the $d$-dimensional measures and the Hausdorff contents are equal to zero, that is, $\mathfrak{m}_d=\mathfrak{m}_d^r=0$ for each $d>2$. As in the preface to [10], an extended scalar function is integrable with respect to the Radon measure on a set if its integral with respect to this measure is correctly defined by a value from $\overline{\mathbb{R}}$. An integrable function is summable if the corresponding integral is finite, that is, takes values from $\mathbb{R}$. If $r\colon S\to \mathbb{R}^+$ is a function on a subset of $S\subset \mathbb{C}$ and $v$ is an $\mathfrak{m}_1$-integrable function on circles $\partial D_z(r(z))$ for $z\in S$, then
$$
\begin{equation}
v^{\circ r}\colon z\underset{z\in S}{\longmapsto}\frac{1}{2\pi} \int_{0}^{2\pi} v\bigl(z+r(z)e^{i\theta}\bigr) \,d \theta
\end{equation}
\tag{2.7}
$$
is the integral average of variable radius $r$ over circles. If a function $v$ is defined on the union of closed discs
$$
\begin{equation}
S^{\cup r}:=\bigcup_{z\in S}\overline D_z(r(z))\subset \mathbb C,
\end{equation}
\tag{2.8}
$$
then its exact upper bounds over the discs $\overline D_z(r(z))$ are denoted by
$$
\begin{equation}
v^{\vee r}\colon z\underset{z\in S}{\longmapsto}\sup_{\overline D_z(r(z))} v;
\end{equation}
\tag{2.9}
$$
for an $\mathfrak m_2$-integrable functions $v$ on these discs, the integral averages of variable radius $r$ over the circles $\overline D_z(r(z))$ is denoted by
$$
\begin{equation}
v^{\bullet r}\colon z\underset{z\in S}{\longmapsto} \frac{1}{\pi(r(z))^2} \int_{\overline D_z(r(z))} v \,d \mathfrak m_2.
\end{equation}
\tag{2.10}
$$
If $u$ is a subharmonic function on an open neighbourhood union of circles (2.8), we have (see Theorem 2.6.8 in [11])
$$
\begin{equation}
u(z)\leqslant u^{\bullet r}(z)\leqslant u^{\circ r}(z) \leqslant u^{\vee r}(z) \quad\text{for all } z\in S^{\cup r}.
\end{equation}
\tag{2.11}
$$
For an $\mathfrak m_1$-measurable subset $E\subset \mathbb{R}^+$, we will use the increasing function (see Lemma 1 in [36])
$$
\begin{equation}
q_E\colon r\underset{r\in \mathbb{R}^+}{\longmapsto} \mathfrak m_1(E\cap [0,r])\ln \frac{er}{\mathfrak m_1(E\cap [0,r])} \underset{r\in \mathbb{R}^+}{\leqslant} r,
\end{equation}
\tag{2.12}
$$
which characterizes this set $E$. The following result solves problems (1.6), (1.7) and (1.8). Main Theorem. Let $\nu$ be a distribution of masses on $\mathbb{C}$ such that Then, for every subharmonic function $M$ of finite type, the following five assertions I– IV are equivalent. I. For any $b\in [0,s)$, there exists a subharmonic function $U\not\equiv -\infty$ of finite type with Riesz distribution of masses $\frac{1}{2\pi}\Delta U\geqslant \nu$ such that
$$
\begin{equation}
U(z)\equiv M(z)\quad\textit{for all } z\in \overline{\operatorname{str}}_b.
\end{equation}
\tag{2.14}
$$
II. If $n\in \mathbb{N}_0$ and $N\in \mathbb{N}$, then
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n<N} \biggl(\ell_{\nu}(2^n,2^N) -\frac{1}{2\pi}\int_{2^n}^{2^N}\frac{M(iy)+M(-iy)}{y^2}\,d y\biggr)<+\infty,
\end{equation}
\tag{2.15}
$$
III. For each pair of subharmonic functions $v$ and $m$ with Riesz distributions of masses, respectively $\frac{1}{2\pi}\Delta v=\nu$ and $\frac{1}{2\pi}\Delta m= \frac{1}{2\pi}\Delta M\lfloor_{\operatorname{str}_{s}}$, and, for any $b\in [0,s)$, there exists an entire function $h\not\equiv 0$ such that the sum $v+m+\ln |h|$ is a subharmonic function of finite type and
$$
\begin{equation}
v(z)+m(z)+\ln|h(z)|\leqslant M(z) \quad\textit{for all } z\in \overline{\operatorname{str}}_b.
\end{equation}
\tag{2.16}
$$
IV. For an arbitrary subharmonic function $v$ with Riesz distribution of masses $\frac{1}{2\pi}\Delta v=\nu$ for any $b\in [0,s)$, $d\in (0,2]$, and a function $r\colon \mathbb{C}\to (0,1]$ such that
$$
\begin{equation}
\inf_{z\in \mathbb{C}} \frac{\ln r(z)}{\ln(2+ |z|)}>-\infty
\end{equation}
\tag{2.17}
$$
there exist an entire function $h\not\equiv 0$ and a subset $E_b\subset \mathbb{C}$ such that $v+\ln |h|$ is a subharmonic function of finite type and
$$
\begin{equation}
v(z)+\ln|h(z)| \leqslant M^{\bullet r}(z) \quad\textit{for all } z\in \overline{\operatorname{str}}_b, \textit{ and also}
\end{equation}
\tag{2.18}
$$
$$
\begin{equation}
v(z)+\ln|h(z)| \leqslant M(z)\quad\textit{for all } z\in \overline{\operatorname{str}}_b\setminus E_b, \textit{ where}
\end{equation}
\tag{2.19}
$$
$$
\begin{equation}
\mathfrak{m}_d^r(E_b\cap S) \leqslant \sup_{z\in S}r(z) \quad\textit{for every }S\subset \mathbb{C}.
\end{equation}
\tag{2.20}
$$
V. There exist an $\mathfrak{m}_1$-measurable function $q_0\colon \mathbb{R}\to \overline{\mathbb{R}}^+$, a continuous function $q\colon \mathbb{R}\to \mathbb{R}^+$ of finite type for which the function $t\underset{t\geqslant t_0}{\longmapsto} (q(t)+q(-t))/t^P$ is decreasing for some $P\in \mathbb{R}^+$ and $t_0\in \mathbb{R}^+$, a subharmonic function of $U\not\equiv -\infty$ of finite type with Riesz distribution of masses $\frac{1}{2\pi}\Delta U\geqslant \nu$, and an $\mathfrak m_1$-measurable set $E\subset \mathbb{R}^+$ with the function $q_E$ from (2.12) such that Remark 1. The conditions on $\operatorname{supp} \nu$ from (2.13) are equivalent to the following single condition. [$ \boldsymbol\nu $] There exist a number $s\in \mathbb{R}^+\setminus 0$ and $a\in (0,1)$ such that Example 1. The power functions $q_0(y)\equiv q(y)\equiv |y|^p$ of degree $p\in [0,1)$ have all the properties required in assertion V. If, at the same time, the order of the function $r\underset{r\in \mathbb{R}^+}{\longmapsto} \mathfrak{m}_1(E\cap [0,r])$ is $<1$, then there exist numbers $C\geqslant 1$, $p_E\in [0,1)$ and $r_E\geqslant 1$ such that $\mathfrak{m}_1(E\cap [0,r])\leqslant Cr^{p_E}\leqslant r$ for all $r\geqslant r_E$. Hence, since the function $x\underset{x\in [0,r]}{\longmapsto} x\ln(er/x)$ is increasing, we have Remark 2. Condition (2.17) for $r\colon \mathbb{C}\to (0,1]$ is equivalent to the existence of a sufficiently small $c\in \mathbb{R}^+\setminus 0$ and large $R\in \mathbb{R}^+$ and $P\in \mathbb{R}^+$ such that Remark 3. The proof of our main theorem follows the scheme 2.2. The case of the distribution of points $\mathrm Z$ in the role of distribution of massesWe consider problem (1.3) in the following Theorem 8, where the distribution of points $\mathrm{Z}\subset \mathbb{C}$ plays the role of the distribution of masses $\nu$ from the main theorem. Theorem 8. Let $M\not\equiv -\infty$ be a subharmonic function of finite type, and let $\mathrm{Z}\subset \mathbb{C}$ be a distribution of points such that, for some $a\in (0,1)$, the external Redheffer density of the restriction $\mathrm{Z}\lfloor_{\overline{\mathrm X}_a}$ along $i\mathbb{R}$ is finite. Then the following four assertions I– IV are equivalent. I. For any $0\leqslant b<s\in \mathbb{R}^+$ and for every subharmonic function $m$ with Riesz distribution of masses $\frac{1}{2\pi}\Delta m=\frac{1}{2\pi}\Delta M\lfloor_{\operatorname{str}_s}$, there exists an entire function $f\not\equiv 0$ vanishing on $\mathrm{Z}$ such that the subharmonic function $\ln |f|+m$ is of finite type and
$$
\begin{equation}
\ln |f(z)|+m(z)\leqslant M(z) \quad\textit{for all } z\in \overline{\operatorname{str}}_b.
\end{equation}
\tag{2.26}
$$
II. If $n\in \mathbb{N}_0$ and $N\in \mathbb{N}$, then
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n<N}\biggl(\ell_\mathrm{Z}(2^n,2^N) - \frac{1}{2\pi}\int_{2^n}^{2^N}\frac{M(iy)+M(-iy)}{y^2}\,d y\biggr)<+\infty.
\end{equation}
\tag{2.27}
$$
III. For any $b\in \mathbb{R}^+$, $d\in (0,2]$ and a function $r\colon \mathbb{C}\to (0,1]$ satisfying (2.17), there exists an entire function $f\not\equiv 0$ of exponential type with $f(\mathrm{Z})=0$ and $E_b\subset \mathbb{C}$ such that $\ln|f(z)|\leqslant M^{\bullet r}(z)$ for all $z\in \overline{\operatorname{str}}_b$, and also $\ln|f(z)|\leqslant M(z)$ for all $z\in \overline{\operatorname{str}}_b\setminus E_b$, where $E_b$ obeys (2.20). IV. For an integer-valued distribution of masses $\nu:=\mathrm{Z}$, assertion V of the main theorem is satisfied with relations (2.21) and (2.22). Proof. By one of the classical Weierstrass theorems, for an arbitrary subharmonic function $v$ with integer-valued distribution of masses $\nu:=\mathrm{Z}$, there exists an entire function $f_\mathrm{Z}\not\equiv 0$ such that
$$
\begin{equation}
\operatorname{Zero}_{f_\mathrm{Z}}=\mathrm{Z}, \qquad v=\ln |f_\mathrm{Z}|.
\end{equation}
\tag{2.28}
$$
Theorem 8 follows from the main theorem by the scheme III $\Rightarrow$ IV $\Rightarrow$ II $\Rightarrow$ I $\Rightarrow$ III.
Let assertion III hold. For the entire function $f\not\equiv 0$, $f(\mathrm{Z})=0$, from assertion III we have the representation $f\stackrel{(2.28)}{=}f_\mathrm{Z}h$. Hence, in view of the equalities $\ln |f|=\ln |f_\mathrm{Z}|+\ln|h|\stackrel{(2.28)}{=}v+\ln|h|$, assertion III gives assertion IV of the main theorem. The implication IV $\Rightarrow$ V of the main theorem is true by Remark 3 without constraints on $\nu$. Therefore, the implication III $\Rightarrow$ IV is also true. Let assertion IV hold. By Remark 3, the implication V $\Rightarrow$ II of the main theorem implies that (2.15) holds with $\nu:=\mathrm{Z}$. The gives us (2.27) with $\mathrm{Z}=\nu$. This proves the implication IV $\Rightarrow$ II. The following result is used in the proof of the implication II $\Rightarrow$ I. Lemma 1. Let $\mathrm{Z}\subset \mathbb{C}$ be an arbitrary distribution of points for which the external Redheffer density along $i\mathbb{R}$ is smaller than some $c\in \mathbb{R}^+\setminus 0$. Then, for any number $s\in \mathbb{R}^+$, there exists an entire function $f_s\not\equiv 0$ of exponential type $\operatorname{type}[\ln|f_s|]<\pi c$ such that $f_s(\mathrm{Z})=0$ and $|f_s(z)|\leqslant 1$ for all $z\in \overline{\operatorname{str}}_{s}$. Proof. The Beurling–Maliavin theorem on the radius of completeness [30], in the variant given by Redheffer in [15], Theorem 2.1.10 and [4], Theorem 2.1.10, after its two reformulations in terms of entire functions of exponential type and a transition via rotation through a right angle from $\mathbb{R}$ to $i\mathbb{R}$, ensures the existence of an entire function $h_s\not\equiv 0$ of exponential type
$$
\begin{equation}
\operatorname{type}[\ln|h_s|]<\pi c, \qquad h_s(\mathrm{Z})=0, \qquad \sup_{y\in \mathbb{R}}|h_s(iy)|\leqslant 1.
\end{equation}
\tag{2.29}
$$
The subharmonic function $\ln|h_s|$ of finite type is bounded on the imaginary axis $i\mathbb{R}$. Hence the Poisson integral of the function $\ln|h_s|$ along the imaginary axis
$$
\begin{equation*}
z\underset{z\in \mathbb{C}\setminus i\mathbb{R}}{\longmapsto} \frac{1}{\pi}\int_{-\infty}^{+\infty} \frac{|{\operatorname{Re} z}|\ln|h_s(iy)|} {(\operatorname{Re} z)^2+(\operatorname{Im} z-y)^2} \,d y+ A|{\operatorname{Re} z}|\stackrel{(2.29)} {\underset{z\in \mathbb{C}\setminus i\mathbb{R}}{\leqslant}} A|{\operatorname{Re} z}|
\end{equation*}
\notag
$$
defines harmonic majorants on $\mathbb{C}_{\mathrm{rh}}$ and on $\mathbb{C}_{\mathrm{lh}}$ for some $A\in \mathbb{R}^+$. So, we have
$$
\begin{equation}
\ln |h_s(z)|\underset{z\in \mathbb{C}}{\leqslant} A|{\operatorname{Re} z}|\leqslant As \quad\text{for all }\ z\in \overline{\operatorname{str}}_s.
\end{equation}
\tag{2.30}
$$
Hence by (2.29) the entire function $f_s:=h_se^{-As}$ of exponential type
$$
\begin{equation*}
\operatorname{type}[\ln|f_s|]= \operatorname{type}[\ln|h_s|]\stackrel{(2.29)}{<}\pi c
\end{equation*}
\notag
$$
vanishes on $Z$, and
$$
\begin{equation*}
\ln|f_s(z)|\leqslant \ln|h_s(z)| -As\stackrel{(2.30)}{\leqslant} As-As=0 \quad\text{for all }z\in \overline{\operatorname{str}}_{s}.
\end{equation*}
\notag
$$
This completes the proof of Lemma 1. Let us derive assertion I from assertion II for $0\leqslant b<s\in \mathbb{R}^+$. The restriction $\mathrm{Z}\lfloor_{{\overline{\mathrm X}_a}\cup \overline{\operatorname{str}}_ s}$ adds to the restriction $\mathrm{Z}\lfloor_{\overline{\mathrm X}_a}$ a finite number of points. Hence, evidently, the distribution of points $\mathrm{Z}\lfloor_{{\overline{\mathrm X}_a}\cup \overline{\operatorname{str}}_ s}$ also has finite external Redheffer density along $i\mathbb{R}$. By Lemma 1 with $\mathrm{Z}\lfloor_{{\overline{\mathrm X}_a}\cup \overline{\operatorname{str}}_ s}$ in place of $\mathrm{Z}$, there exists an entire function $f_s$ of exponential type such that
$$
\begin{equation}
f_s(\mathrm{Z}\lfloor_{{\overline{\mathrm X}_a}\cup \overline{\operatorname{str}}_ s})=0, \qquad |f_s(z)|\leqslant 1 \quad\text{for all }z\in \overline{\operatorname{str}}_{s}.
\end{equation}
\tag{2.31}
$$
In parallel with the the distribution of points $\mathrm{Z}\lfloor_{{\overline{\mathrm X}_a}\cup \overline{\operatorname{str}}_ s}$, we consider the remainder part of the distribution of points $\mathrm Z$. This distribution of points, when regarded as an integer-valued measure $\nu:=\mathrm{Z}\lfloor_{\mathbb{C}\setminus ({\overline{\mathrm X}_a} \cup \overline{\operatorname{str}}_ s)}\leqslant \mathrm{Z}$, clearly satisfies condition [$ \boldsymbol\nu $] with (2.23) from Remark 1, which is equivalent to conditions (2.13) of the main theorem. Inequality (2.27) follows from (2.15). For an integer-valued distribution of masses $\nu$, there exists an entire function $F\not\equiv 0$ with distribution of zeros $\operatorname{Zero}_F= \mathrm{Z}\lfloor_{\mathbb{C}\setminus ({\overline{\mathrm X}_a}\cup \overline{\operatorname{str}}_ s)}\leqslant \mathrm{Z}$. In other words, the function $v:=\ln|F|$ is subharmonic with Riesz distribution mass $\nu$. By the implication II $\Rightarrow$ III of the main theorem, for this subharmonic function $v=\ln|F|$, there exists an entire function $h\not\equiv 0$ such that is a subharmonic function of finite type and (2.16) holds. Hence
$$
\begin{equation}
\ln|F(z)h(z)|+m(z) \stackrel{(2.32)} {\underset{z\in \mathbb{C}}{\equiv}} v(z)+m(z)+\ln|h(z)| \underset{z\in \overline{\operatorname{str}}_b} {\stackrel{(2.16)}{\leqslant}} M(z),
\end{equation}
\tag{2.33}
$$
where the entire function $Fh$ vanishes on $\mathrm{Z}\lfloor_{\mathbb{C}\setminus ({\overline{\mathrm X}_a}\cup \overline{\operatorname{str}}_ s)}=\operatorname{Zero}_F$. Hence, for the entire function $f:=f_sFh$, the function $\ln|f|+m=\ln|f_s|+\ln |Fh|+m$ is a subharmonic function of finite type (as a sum of such functions). At the same time,
$$
\begin{equation*}
\ln|f(z)|+m(z)=\ln|f_s(z)|+\ln |F(z)h(z)|+m(z) \stackrel{(2.31),(2.33)} {\underset{z\in \overline{\operatorname{str}}_b}{\leqslant}} M(z),
\end{equation*}
\notag
$$
where $f\not\equiv 0$ vanishes on $\mathrm{Z}$, since for the multiplier $f_s$ we have (2.31), and the multiplier $Fh$ vanishes on $\mathrm{Z}\lfloor_{\mathbb{C}\setminus (\overline{\mathrm X}_a\cup \overline{\operatorname{str}}_ s)}$. Thus, we have shown that assertion II implies assertion I. Finally, let assertion I hold. To derive assertion III from assertion I, for a given $b\geqslant 0$, we choose $s>b$ and again consider an arbitrary subharmonic function $v$ with an integer-valued Riesz distribution of masses $\frac{1}{2\pi}\Delta v:=\nu:=\mathrm{Z}$, as in (2.28). The entire function $f\not\equiv 0$ from assertion I is represented as $f\stackrel{(2.28)}{=}f_\mathrm{Z}h$, where $h\not\equiv 0$ is the entire function. Moreover, $v+m+\ln |h|=\ln |f_\mathrm{Z}h|+m=\ln |f|+m$. Hence, by assertion I, the subharmonic function $v+m+\ln |h|$ is of finite type, and from (2.26) we obtain
$$
\begin{equation*}
v(z)+m(z)+\ln |h(z)|\underset{z\in \mathbb{C}}{\equiv} \ln |f(z)|+m(z) \stackrel{(2.26)}{\leqslant} M(z) \quad\text{for all } z\in \overline{\operatorname{str}}_b.
\end{equation*}
\notag
$$
This means that assertion III of the main theorem holds for $\nu=\mathrm{Z}$. The implication III $\Rightarrow$ IV of the main theorem holds by Remark 3 without any constraint on $\nu$. Therefore, for the function $v=\ln|f_\mathrm{Z}|$, there exists an entire function $h\not\equiv 0$ satisfying inequalities (2.18)–(2.20) with the left-hand side $v+\ln|h|=\ln |f_\mathrm{Z}h|$, which is a subharmonic function of finite type. This means that, for the distribution of masses $\mathrm{Z}=\nu$, the entire function $f:=f_\mathrm{Z}h$ vanishes on $\mathrm{Z}$ and has the properties required in assertion III. Thus, the implication I $\Rightarrow$ III holds. This completes the proof of Theorem 8. The following Corollary 1 allows us to combine assertions I and III of Theorem 8 into a single simpler result, and completely get rid of both the integral averaging $M^{\bullet}$ and the exceptional set $E_b$ in assertion III of Theorem 8. Corollary 1. If, under the conditions of Theorem 8, for some $s\in \mathbb{R}^+\setminus 0$, the restriction $\frac{1}{2\pi}\Delta M\lfloor_{\operatorname{str}_s}$ is an integer-valued distribution of masses, then the assertions I and III of Theorem 8 can be replaced by the following single stronger assertion, I $\cap$ III. For any $b\in [0,s)$, there exists an entire function $f\not\equiv 0$ of exponential type such that $f(\mathrm{Z})=0$ and $\ln |f(z)|\leqslant M(z)$ for all $z\in \overline{\operatorname{str}}_b$. This assertion is equivalent to assertions II and IV of Theorem 8. Proof. For any subharmonic function of $m$ with integer-valued distribution of the masses $\frac{1}{2\pi}\Delta m=\frac{1}{2\pi}\Delta M\lfloor_{\operatorname{str}_s}$ by assertion I of Theorem 8, there exists an entire function $l\not\equiv 0$ such that, for $m=\ln |l|$, the function $\ln |fl|=\ln|f|+\ln|l|=\ln|f|+m$ is a subharmonic function of finite type such that
$$
\begin{equation}
\ln |f(z)l(z)|\underset{z\in\mathbb{C}}{\stackrel{(2.26)}{\equiv}} \ln |f(z)|+m(z)\leqslant M(z) \quad\text{for all } z\in \overline{\operatorname{str}}_b.
\end{equation}
\tag{2.34}
$$
Thus, (2.34) is fulfilled for this entire function $fl$ of exponential type. If the product $fl$ is considered as an entire function $f$ of exponential type, then assertion I of Theorem 8 is precisely assertion I $\cap$ III, which is obviously stronger than assertion III of Theorem 8. Corollary 2. For any entire function $g\not\equiv 0$ of exponential type and the same distribution of points ${\mathrm Z}$ as in Theorem 8, either of assertions I and II of Theorem 6 is equivalent to assertion IV of Theorem 8. Proof. Let $M:=\ln |g|\not\equiv -\infty$. Then, for any number $s>0$, the restriction $\frac{1}{2\pi}\Delta M\lfloor_{\operatorname{str}_s}= \operatorname{Zero}_g\lfloor_{\operatorname{str}_s}$ is an integer-valued distribution of masses, and assertions I and II of Theorem 6 are precisely assertion I $\cap$ III from Corollary 1 and assertion II of Theorem 8. Thus, Corollary 1 implies the Corollary 2. Proof of Theorem 6. Assertion I with $E:=\varnothing$ obviously implies assertion III, which by Example 1 is a special case of assertion IV of Theorem 8. Thus, Corollary 2 entails Theorem 6. Proof of Theorem 4. The equivalence II $\Leftrightarrow$ III is a dual version of the Beurling–Malliavin theorem on the radius of completeness in combination with the classical Paley–Wiener theorem, and the equivalence III $\Leftrightarrow$ IV is one of the forms of the Beurling–Malliavin theorem on multipliers. The implication I $\Rightarrow$ III is obvious when the choice $g\equiv 1$. Finally, if assertion II holds, then by Lemma 1, for any $s\in \mathbb{R}^+$, there exists an entire function $f_s\not\equiv 0$ of exponential type $\operatorname{type}[\ln|f_s|]<\pi c$ such that $f_s(\mathrm{Z})=0$ and $|f_s(z)|\leqslant 1$ for all $z\in \overline{\operatorname{str}}_{s}$. Hence, for an arbitrary entire function $g\not\equiv 0$ of exponential type, the entire function $f:=f_sg\not\equiv 0$ of exponential type
$$
\begin{equation*}
\operatorname{type} [\ln|f|]\leqslant \operatorname{type} [\ln|f_s|]+\operatorname{type} [\ln|g|] <c+\operatorname{type} [\ln|g|]
\end{equation*}
\notag
$$
vanishes on $\mathrm{Z}\leqslant \operatorname{Zero}_{f_s}\leqslant \operatorname{Zero}_f$ and satisfies
$$
\begin{equation*}
\ln|f(z)|\equiv \ln|f_s(z)|+\ln|g(z)|\leqslant \ln|g(z)|\quad\text{for all } z\in \overline{\operatorname{str}}_s.
\end{equation*}
\notag
$$
This completes the proof of Theorem 4. 2.3. Proofs of the implications preceding II in (2.25) Proof of the implication IV $\Rightarrow$ V. By assertion IV with $b:=0$, there exists a subharmonic function $U:=v+\ln|h|\not\equiv -\infty$ of finite type in (2.19)–(2.20) with Riesz distribution of masses $\frac{1}{2\pi}\Delta U\geqslant \nu$ which satisfies $U(iy)\leqslant M(iy)$ for all $iy\in (\mathbb{C}\setminus E_0)\cap i\mathbb{R}$, where, for $d:=1$ and with the choice of $r\equiv 1$ in (2.17), we have $\mathfrak{m}_1^1(E_0) <+\infty$. By the definition of the Hausdorff content (2.4) of radius $1$, the last inequality means that some union of closed intervals on $i\mathbb{R}$ with finite sum of lengths covers $iE':=E_0\cap i\mathbb{R}$. By assertion IV,
$$
\begin{equation}
U(iy)\leqslant M(iy)\quad\text{for all }y\in \mathbb{R}\setminus E', \text{ where }\mathfrak{m}_1(E') <+\infty.
\end{equation}
\tag{2.35}
$$
Hence, for $E:=(E'\cup (-E'))\cap \mathbb{R}^+$ we have, as before, $\mathfrak{m}_1(E) <+\infty$, and
$$
\begin{equation}
U(iy)+U(-iy)\leqslant M(iy)+M(-iy)\quad\text{for all } y\in \mathbb{R}^+\setminus E.
\end{equation}
\tag{2.36}
$$
For this subset $E\,{\subset}\,\mathbb{R}^+$, by definition of the function $q_E$ (see (2.12)) for $r\geqslant \mathfrak{m}_1(E)$ we have
$$
\begin{equation}
q_{E}(r) \stackrel{(2.12)}{\leqslant} \mathfrak{m}_1(E) \ln\frac{er}{\mathfrak{m}_1(E)}\underset{r\to +\infty}{=}O(\ln r)
\end{equation}
\tag{2.37}
$$
since the function $x \mapsto x\ln(er/x)$, $x\in [0,r]$, is increasing. Hence, choosing $q_0=q=0$, we prove that the integral in (2.22) finite, and, verify, in view of (2.36), inequality (2.21) for all $y\in \mathbb{R}^+\setminus E$. Remark 4. The proofs of the implications I $\Rightarrow$ V and III $\Rightarrow$ V are even simpler, since the choice of a function $U$ in the first case is the same as in assertion I, in the second case, we can put $U:=v+m+\ln|h|$, $E:=\varnothing$ for both cases, and the argument following from (2.36) is the same as in the proof of the implication IV $\Rightarrow$ V. Moreover, the main theorem can be supplemented with the following single assertion, which is equivalent to assertions I–V, but which is simpler and more laconic than assertion V. VI. There is a subharmonic function $U\not\equiv -\infty$ of finite type, $\frac{1}{2\pi}\Delta U\geqslant \nu$, and an $\mathfrak m_1$-measurable subset $E'\subset \mathbb{R}$ such that
$$
\begin{equation}
U(iy)\leqslant M(iy)\quad\textit{for all }y\in \mathbb{R}\setminus E', \textit{ where } \mathfrak{m}_1(E')<+\infty.
\end{equation}
\tag{2.38}
$$
Indeed, the proof of the implication IV $\Rightarrow$ VI is a verbatim repetition of that for the implication IV $\Rightarrow$ V (up to (2.35) inclusively), and the proof of the implication VI $\Rightarrow$ V is the same as for the implication IV $\Rightarrow$ V from (2.35) to the end of the proof. Proof of the implication V $\Rightarrow$ II. Assertion V up to (2.21) inclusively is precisely the assumption of the main Theorem 1 in [36], the only difference is that it does not even assume that the function
$$
\begin{equation}
t\underset{t\geqslant t_0}{\longmapsto} \frac{q(t)+q(-t)}{t^P} \quad\text{for some }P\in \mathbb{R}^+ \text{ and }t_0\in \mathbb{R}^+,
\end{equation}
\tag{2.39}
$$
should decrease, where, by increasing $P$ if necessary, it can be assumed that $P\geqslant 2$, keeping the function (2.39) decreasing. Now, by the substantially relaxed conclusion from [36], Theorem 1, formulas (1.10), (1.11), for any numbers $r_0>0$ and $N\in \mathbb{R}^+$, there exists $C\in \mathbb{R}^+$ such that
$$
\begin{equation}
\begin{aligned} \, &\ell_{\nu}(r,R)\leqslant \frac{1}{2\pi}\int_{r}^{R}\frac{M(iy)+M(-iy)}{y^2}\,d y +C\frac{1}{2\pi}\int_{r}^{R}\frac{q_0(t)+q_0(-t)+2q_E(t)}{t^2}\,d t \nonumber \\ &\qquad+C \int_r^R t^N\sup_{s\geqslant t} \frac{q(s)+q(-s)}{s^{2+N}}\,d t+C \quad\text{for all }r_0\leqslant r<R<+\infty. \end{aligned}
\end{equation}
\tag{2.40}
$$
We can put here $N:=P-2\in \mathbb{R}^+$, since $P\geqslant 2$. Under the assumption that function (2.39) decreases, and for $P=2+N$, for some sufficiently large $t_0\in \mathbb{R}^+$, there exist (see Proposition 1 in [36]) numbers $r_1> 0$ and $C_1\in \mathbb{R}^+$ such that the last integral in (2.40) is estimated from above
$$
\begin{equation*}
\int_r^R t^N\sup_{s\geqslant t} \frac{q(s)+q(-s)}{s^{2+N}}\,d t\leqslant C_1\int_r^R \frac{q(t)+q(-t)}{t^2}\,d t \quad\text{for all }r_1\leqslant r<R<+\infty.
\end{equation*}
\notag
$$
The last estimate together with (2.40) gives (see [36], Proposition 1, formula (3.4))
$$
\begin{equation*}
\begin{aligned} \, \ell_{\nu}(r,R) &\leqslant \frac{1}{2\pi}\int_{r}^{R}\frac{M(iy)+M(-iy)}{y^2}\,d y \\ &\qquad+C_2\frac{1}{2\pi}\int_{r}^{R}\frac{q_0(t)+ q_0(-t)+2q_E(t)+q(t)+q(-t)}{t^2}\,d t +C_2 \end{aligned}
\end{equation*}
\notag
$$
for some $C_2\in \mathbb{R}^+$ for all $r_2:=\max\{1, r_0, r_1\}\leqslant r<R<+\infty$, whence by (2.22) for some $C_3\in \mathbb{R}$ we have (see Corollary 1 in [36])
$$
\begin{equation*}
\ell_{\nu}(r,R)\leqslant \frac{1}{2\pi}\int_{r}^{R}\frac{M(iy)+M(-iy)}{y^2}\,d y +C_3\quad\text{for all }r_2\leqslant r<R<+\infty.
\end{equation*}
\notag
$$
Here, the fixed value $r_2$ can be replaced by $1$ (possibly increasing $C_3$) since the subharmonic functions are locally ${\mathfrak m}_1$-summable on each line and in view of the subadditivity inequality $\ell_{\nu}(r,R)\stackrel{(2.3)}{\leqslant} \ell_{\nu}(1,r_2)+\ell_{\nu}(r_2,R)$ for $r\in [1,r_2)$ (see § 3.2 in [36]). After this, transferring the integral from the right-hand side with the opposite sign to the left-hand side of the inequality and applying the operation $\sup_{r\in [1,R)}$, and then $\limsup_{R\to +\infty}$ to the resulting difference on the left, we get
$$
\begin{equation}
\limsup_{R\to +\infty} \sup_{1\leqslant r<R}\biggl(\ell_{\nu}(r,R)- \frac{1}{2\pi}\int_{r}^{R}\frac{M(iy)+M(-iy)}{y^2}\,d y\biggr)<+\infty.
\end{equation}
\tag{2.41}
$$
Here, the left-hand side is majorized by that in (2.15). This proves assertion II. Remark 5. The proof of the key implication II $\Rightarrow$ I in the main theorem, which will be given in § 7, requires a substantial preliminary preparation in §§ 3–6 based on the balayage machinery. The proofs of the remaining implications I $\Rightarrow$ III $\Rightarrow$ IV, which are given in § 9, are based on Theorem 9 from § 8. This result shows, that, for an arbitrary subharmonic function $u\not\equiv -\infty$, there exists an entire function $f\not\equiv 0$ such that $\ln |f|\leqslant u$ outside a very small exceptional set. Theorem 9 is also of independent interest. § 3. Balayages from the right half-plane3.1. Balayage of genus $0$ and $1$ of distribution of chargesFor a distribution of charges $\nu$, we use (see formula (1.9) in [9]) its distribution function $\nu_{\mathbb{R}}\colon \mathbb{R}\to \mathbb{R}$ defined by
$$
\begin{equation}
\nu_{\mathbb{R}}(x_2)-\nu_{\mathbb{R}}(x_1):=\nu((x_1,x_2]), \qquad -\infty <x_1<x_2<+\infty,
\end{equation}
\tag{3.1}
$$
and its distribution function $\nu_{i\mathbb{R}}\colon \mathbb{R}\to \mathbb{R}$ on $i\mathbb{R}$ is defined by
$$
\begin{equation}
\nu_{i\mathbb{R}}(y_2)-\nu_{i\mathbb{R}}(y_1):=\nu(i(y_1,y_2]), \qquad -\infty <y_1<y_2<+\infty.
\end{equation}
\tag{3.2}
$$
These distribution functions are defined only up to an additive constant; we can normalize them, if necessary, at zero
$$
\begin{equation}
\nu_{\mathbb{R}}(0):=0, \qquad \nu_{i\mathbb{R}}(0):=0.
\end{equation}
\tag{3.3}
$$
By construction (see (3.1) and (3.2)), the functions $\nu_{\mathbb{R}}$ and $\nu_{i\mathbb{R}}$ are of locally bounded variation on $\mathbb{R}$, and, in the case of distributions of masses $\nu$, both these functions are increasing. Conversely, any function of locally bounded variation on $\mathbb{R}$ or on $i\mathbb{R}$ uniquely defines the distribution of charges supported on $\mathbb{R}$ or $i\mathbb{R}$, respectively. We recall and adapt the basic concepts and results from [9] and [28], and also partially from [48] and [35] on the balayage of finite genus $q\in \mathbb{N}_0$ of distributions of charges, but for now, only in the context of the right half-plane $\mathbb{C}_{\mathrm{rh}}$ in the cases $q:=0$ and $q:=1$. The papers [9] and [28] are mainly concerned with the upper half-plane $\mathbb{C}^{\mathrm{up}}:=i\mathbb{C}_{\mathrm{rh}}$; this setting can be carried over to $\mathbb{C}_{\mathrm{rh}}$ by rotating through a right angle. Let
$$
\begin{equation}
\boldsymbol{1}_S\colon z\underset{z\in \mathbb C}{\longmapsto} \begin{cases} 1 &\text{if }z\in S, \\ 0 &\text{if }z\notin S \end{cases}
\end{equation}
\tag{3.4}
$$
be the characteristic function of a set $S$. The harmonic measure for $\mathbb{C}_{\mathrm{rh}}$ at $z\in \mathbb{C}_{\mathrm{rh}}$ on intervals $i(y_1,y_2]\subset i\overline{\mathbb{R}}$
$$
\begin{equation}
\omega_{\mathrm{rh}} \bigl(z,i(y_1,y_2]\bigr) \stackrel{\text{[9, 3.1]}}{:=} \omega_{\mathbb{C}_{\mathrm{rh}}}(z,i(y_1,y_2]) \underset{z\in \mathbb{C}_{\mathrm{rh}}}{:=} \frac1{\pi} \int_{y_1}^{y_2}\operatorname{Re} \frac{1}{z-iy} \,d y
\end{equation}
\tag{3.5}
$$
is equal to the angle divided by $\pi$ under which the interval $i(y_1,y_2]$ is visible from the point $z\in \mathbb{C}_{\mathrm{rh}}$ (see formula (3.1)) in [35] and §§ 1.2.1, 3.1 in [9]), and, at points of the imaginary axis $i\mathbb{R}$, this measure is defined by
$$
\begin{equation}
\omega_{\mathrm{rh}} \bigl(iy,i(y_1,y_2]\bigr):=\boldsymbol{1}_{(y_1,y_2]}(y) \quad\text{at }y\in \mathbb{R}.
\end{equation}
\tag{3.6}
$$
Let $\nu $ be a distribution of charges satisfiyng the classical Blaschke condition for $\mathbb{C}_{\mathrm{rh}}$
$$
\begin{equation}
\ell_{|\nu|}^{\operatorname{rh}}(1,+\infty)\stackrel{(2.3)}{=} \int_{\mathbb{C}_{\mathrm{rh}}\setminus \mathbb{D}} \operatorname{Re} \frac{1}{z}\,d |\nu| (z)<+\infty.
\end{equation}
\tag{3.7}
$$
For $\nu$, its classical balayage from $\mathbb{C}_{\mathrm{rh}}$ to $ \mathbb{C}_{\overline{\mathrm{lh}}}$ with support on $ \mathbb{C}_{\overline{\mathrm{lh}}}$ is correctly defined (see [9], Corollary 4.1, Theorem 4]), which in a broader framework (see Definition 3.1 in [28]) represents the balayage of genus $0$ (denoted in [28] by $\nu^{\operatorname{bal}[0]}_{\mathbb{C}_{\overline{\mathrm{lh}}}}$). Here, we use a slightly more compact notation $\nu^{\operatorname{bal}^0_{\mathrm{rh}}}:= \nu^{\operatorname{bal}[0]}_{\mathbb{C}_{\overline{\mathrm{lh}}}}$. By definition, the distribution of charges $\nu^{\operatorname{bal}^0_{\mathrm{rh}}}$ is the sum of the restriction $\nu\lfloor_{\mathbb{C}_{\mathrm{lh}}}$ to $\mathbb{C}_{\mathrm{lh}}$ with the distribution of charges on $i\mathbb{R}$ defined (in notation (3.2)) by the distribution function
$$
\begin{equation}
\nu^{\operatorname{bal}^0_{\mathrm{rh}}}_{i\mathbb{R}}(y_2) - \nu^{\operatorname{bal}^0_{\mathrm{rh}}}_{i\mathbb{R}}(y_1) \stackrel{(3.5),(3.6)}{:=} \int_{\mathbb{C}_{\overline{\mathrm{rh}}}} \omega_{\mathrm{rh}}\bigl(z, i(y_1,y_2]\bigr) \,d \nu(z)
\end{equation}
\tag{3.8}
$$
(with normalization (3.3) if necessary). A classical balayage of genus $0$ does not increase the full measure of total variation of the charge distribution, since the harmonic measure (3.5) is probabilistic, and
$$
\begin{equation}
\bigl|\nu^{\operatorname{bal}^0_{\mathrm{rh}}}\bigr|(S) \underset{S\subset \mathbb{C}}{\stackrel{(3.8)}{\leqslant}} |\nu|(S).
\end{equation}
\tag{3.9}
$$
The definition of a harmonic charge of genus $1$ for the upper half-plane $\mathbb{C}^{\mathrm{up}}$ at a point $z\in \mathbb{C}^{\mathrm{up}}$ was defined in Definition 2.1 of [28], and was denoted there by $\Omega^{[1]}_{\mathbb{C}^{\mathrm{up}}}$. Here, we use the right angle rotation with transition from $\mathbb{C}^{\mathrm{up}}$ to $\mathbb{C}_{\mathrm{rh}}$, and define the harmonic charge of genus $1$ for the right half-plane $\mathbb{C}_{\mathrm{rh}}$ as a function $\Omega_{\mathrm{rh}}$ of bounded intervals $i(y_1,y_2]\subset i\mathbb{R}$ by the rule
$$
\begin{equation}
\Omega_{\mathrm{rh}}\bigl(z,i(y_1,y_2]\bigr) \stackrel{(3.5),(3.6)}{:=} \omega_{\mathrm{rh}}\bigl(z,i(y_1,y_2]\bigr)- \frac{y_2-y_1}{\pi}\operatorname{Re}\frac{1}{z}\quad\text{in } z\in \mathbb{C}_{\overline{\mathrm{rh}}}\setminus 0.
\end{equation}
\tag{3.10}
$$
A balayage $\nu_{\mathbb{C}_{\overline{\mathrm{lh}}}}^{\operatorname{bal}[1]}$ of genus $1$ of a distribution of charges $\nu$ from $\mathbb{C}_{\mathrm{rh}}$ onto $ \mathbb{C}_{\overline{\mathrm{lh}}}$, provided $0\notin \operatorname{supp} \nu$, was determined in [28], Definition 3.1, Theorem 1, Remark 3.3. Here, this balayage is denoted more compactly as $\nu^{\operatorname{bal}^1_{\mathrm{rh}}} := \nu^{\operatorname{bal}[1]}_{\mathbb{C}_{\overline{\mathrm{lh}}}}$. By definition, the distribution of charges $\nu^{\operatorname{bal}^1_{\mathrm{rh}}}$ is the sum of the restriction $\nu\lfloor_{\mathbb{C}_{\mathrm{lh}}}$ with the distribution of charges on $i\mathbb{R}$ defined (in notation (3.2)) by the distribution function
$$
\begin{equation}
\nu^{\operatorname{bal}^1_{\mathrm{rh}}}_{i\mathbb{R}}(y_2) - \nu^{\operatorname{bal}^1_{\mathrm{rh}}}_{i\mathbb{R}}(y_1) \stackrel{(3.10)}{=}\int_{\mathbb{C}_{\mathrm{rh}}} \Omega_{\mathrm{rh}} \bigl(z, i(y_1,y_2]\bigr)\,d \nu (z)
\end{equation}
\tag{3.11}
$$
with normalization (3.3) if necessary. Remark 6. A balayage of the distribution of charges of genus $1$ from $\mathbb{C}_{\mathrm{rh}}$ onto $\mathbb{C}_{\overline{\mathrm{lh}}}$ is a part of a global (or two-sided) balayage of the distribution of charges from $\mathbb{C}\setminus i\mathbb{R}$ onto $i\mathbb{R}$, which played a key role in § 3 of [35]. A bilateral balayage onto the imaginary axis can be considered as a sequential balayage of genus $1$ of a charge distribution, first, from $\mathbb{C}_{\mathrm{rh}}$ onto $ \mathbb{C}_{\overline{\mathrm{lh}}}$, and then, by the reflection symmetric balayage procedure of genus $1$ with respect to $i\mathbb{R}$ of the resulting distribution of masses from $\mathbb{C}_{\mathrm{lh}}$ onto $\mathbb{C}_{\overline{\mathrm{rh}}}$. The restriction $0\notin \operatorname{supp} \nu$ for a balayage of genus $1$ can be easily overcome by combining a balayage of genus $0$ of the part of $\nu$ near the origin with a balayage of genus $1$ for the remaining part of $\nu$. To do this, we define the combined balayage of genus $01$ of the distribution of charges $\nu$ from $\mathbb{C}_{\mathrm{rh}}$ onto $\mathbb{C}_{\overline{\mathrm{lh}}}$ (see [28], Remarks 3.3, formulas (3.43), (4.1))
$$
\begin{equation}
\nu^{\operatorname{bal}_{\mathrm{rh}}^{01}}:= \bigl(\nu\lfloor_{r_0\mathbb{D}}\bigr)^{\operatorname{bal}_{\mathrm{rh}}^{0}} +\bigl(\nu\lfloor_{\mathbb{C}\setminus r_0\mathbb{D}}\bigr)^{\operatorname{bal}_{\mathrm{rh}}^{1}}
\end{equation}
\tag{3.12}
$$
for some fixed radius $r_0\in \mathbb{R}^+\setminus 0$. Remark 7. The disc $r_0\mathbb{D}$ on the right of (3.12) can be replaced by any bounded Borel set containing the semidisc $r_0\mathbb{D}\setminus \mathbb{C}_{\overline{\mathrm{lh}}}$ (or even by the empty set). In other words, we can proceed entirely without any balayage of genus $0$ and define $\nu^{\operatorname{bal}_{\mathrm{rh}}^{01}}:= \nu^{\operatorname{bal}_{\mathrm{rh}}^{1}}$ if $0\notin \operatorname{supp} \nu$ or, more generally, which is obviously true if, for some $r_0\in \mathbb{R}^+\setminus 0$, we have Now the question of the existence of balayage $\nu^{\operatorname{bal}_{\mathrm{rh}}^{01}}$ depends only on the behaviour of the charge distribution $\nu$ near infinity. One case here is simple. Proposition 1. If a compactly supported distribution of charges $\nu$ is concentrated in $\mathbb{C}_{\mathrm{rh}}$ and satisfies (3.13) or (3.14), then its balayage $\nu^{\operatorname{bal}^1_{\mathrm{rh}}}$ of genus $1$ is related with the balayage $\nu^{\operatorname{bal}^0_{\mathrm{rh}}}$ of genus $0$ via the distribution function by Proof. The support being compact, the balayage $\nu^{\operatorname{bal}^0_{\mathrm{rh}}}$ of genus $0$ of the distribution of charges $\nu$ with distribution function (3.8) is correctly defined. By Remark 7, the last integral in (3.15) exists. Hence from the form (3.10) of the harmonic charge $\Omega_{\mathrm{rh}}$ and definitions (3.8) and (3.11) we get (3.15), from which the integral equality (3.16) readily follows. This proves Proposition 1. The distribution of charges $\nu$ lies in the convergence class of growth order $p\in \mathbb{N}_0$ if (see Definition 4.1 in [12] and § 2, 2.1, (2.3) in [9]
$$
\begin{equation}
\int_1^{+\infty}\frac{|\nu|^{\mathrm{rad}}(t)}{t^{p+1}}\,d t<+\infty.
\end{equation}
\tag{3.17}
$$
Note that in [38], § 3, various different kinds of the Lindelöf conditions for distribution of charges $\nu$ were used. The distribution of charges $\nu$ satisfies the Lindelöf $\mathbb{R}$-condition (of genus 1) if
$$
\begin{equation}
\sup_{r\geqslant 1} \biggl| \int_{1<|z|\leqslant r}\operatorname{Re} \frac{1}{z}\,d \nu(z)\biggr|<+\infty,
\end{equation}
\tag{3.18}
$$
which by definition (2.1), (2.2) is equivalent to saying that
$$
\begin{equation}
\sup_{r\geqslant 1} \bigl| \ell^{\operatorname{rh}}_{\nu}(1,r) - \ell^{\operatorname{lh}}_{\nu}(1,r)\bigr|<+\infty.
\end{equation}
\tag{3.19}
$$
The distribution of charges $\nu$ satisfies the Lindelöf $i\mathbb{R}$-condition (of genus $1$) if
$$
\begin{equation}
\sup_{r\geqslant 1}\biggl| \int_{1<|z|\leqslant r}\operatorname{Im} \frac{1}{z}\,d \nu(z) \biggr|<+\infty.
\end{equation}
\tag{3.20}
$$
The distribution of charges $\nu$ satisfies the Lindelöf condition (of genus $1$) if
$$
\begin{equation}
\sup_{r\geqslant 1}\biggl|\int_{1<|z|\leqslant r}\frac{1}{z}\,d \nu(z) \biggr|<+\infty.
\end{equation}
\tag{3.21}
$$
The key role of the Lindelöf conditions is reflected in the following classical result. Theorem (Weierstrass–Hadamard–Lindelöf–Brelot theorem; see [12], §§ 4.1, 4.2, [49], § 33, Theorem 12, [14], § 2.9.3, and [9], § 6.1). If $u\not\equiv -\infty$ is a subharmonic function of finite type, then its Riesz distribution of masses $\frac{1}{2\pi}\Delta u$ is of finite upper density and satisfies the Lindelöf condition (3.21). Conversely, if a distribution of masses $\nu$ is of finite upper density, then there exists a subharmonic function $u_{\nu}$ with $\frac{1}{2\pi}\Delta u_{\nu}=\nu$ of order $\operatorname{ord}[u_{\nu}]\leqslant 1$ such that the function $u_{\nu}$ is of finite type under the Lindelöf condition (3.21) for $\nu$. Besides, every subharmonic function $u$ with $\frac{1}{2\pi}\Delta u=\nu$ can be represented as the sum $u=u_{\nu}+H$, where $H$ is a harmonic function on $\mathbb{C}$, which, under the condition $\operatorname{type}_2[u]=0$, is a harmonic polynomial of degree $\deg H\leqslant 1$, and the function $u$ is a function of order $\operatorname{ord}[u]\leqslant 1$. Proposition 2. Let $\nu$ be a distribution of charges such that the restriction $\nu\lfloor_{\mathbb{C}_{\mathrm{rh}}}$ belongs to the convergence class for order $p\stackrel{(3.17)}{=}2$. Then there exists a balayage $\nu^{\operatorname{bal}^{01}_{\mathrm{rh}}}$ of genus $01$ from $\mathbb{C}_{\mathrm{rh}}$ onto $\mathbb{C}_{\overline{\mathrm{lh}}}$. In particular, if $\operatorname{ord}[\nu]<2$, then $\nu$ is from the convergence class for order $<2$ and $\operatorname{ord}[\nu^{\operatorname{bal}^{01}_{\mathrm{rh}}}]\leqslant \operatorname{ord}[\nu]$. If $\nu$ is of finite upper density and satisfies the condition
$$
\begin{equation}
\sup_{r\geqslant 1}\bigl|\ell_{\nu}^{\operatorname{rh}}(1,r)\bigr| \stackrel{(2.1)}{<}+\infty,
\end{equation}
\tag{3.22}
$$
then $\nu^{\operatorname{bal}^{01}_{\mathrm{rh}}}$ is the distribution of charges of finite upper density, and the difference $\nu-\nu^{\operatorname{bal}^{01}_{\mathrm{rh}}}$, satisfies the Lindelöf condition. In addition, if the support $\operatorname{supp} \nu$ is disjoint from a closed angle of opening $>\pi$ containing $i\mathbb{R}$, with vertex at zero and the bisector $-\mathbb{R}^+$, that is,
$$
\begin{equation}
\bigl\{z\in \mathbb{C} \bigm| \operatorname{Re} z\leqslant a|z|\bigr\} \cap \operatorname{supp} \nu=\varnothing\quad\textit{for some } a\in (0,1),
\end{equation}
\tag{3.23}
$$
then, in notation (1.13), where $\bigl|\nu^{\operatorname{bal}^{01}_{\mathrm{rh}}}\bigr|_{iy}^{\mathrm{rad}}$ is the radial counting function with centre at $iy\in i\mathbb{R}$ for the total variation $\bigl|\nu^{\operatorname{bal}^{01}_{\mathrm{rh}}}\bigr|$ of the distribution of charges $\nu^{\operatorname{bal}^{01}_{\mathrm{rh}}}$. Proof. For the classical balayage $(\nu\lfloor_{r_0\overline{\mathbb{D}}})^{\operatorname{bal}_{\mathrm{rh}}^{0}}$ of genus $0$ from the right-hand side of (3.12), we have
$$
\begin{equation*}
\bigl|(\nu\lfloor_{r_0\mathbb{D}})^{\operatorname{bal}_{\mathrm{rh}}^{0}} \bigr|(\mathbb{C}) \stackrel{(3.9)}{\leqslant} |\nu|(r_0\mathbb{D})
\end{equation*}
\notag
$$
Obviously, $(\nu\lfloor_{r_0\mathbb{D}})^{\operatorname{bal}_{\mathrm{rh}}^{0}}$ has a finite upper density and the Lindelöf condition is satisfied. Therefore, we can further proceed only with the distribution of charges $\nu$ such that
$$
\begin{equation}
\operatorname{supp} \nu \subset \mathbb{C}\setminus 2d\overline{\mathbb{D}} \quad\text{for some } d\in (0,1]
\end{equation}
\tag{3.25}
$$
and consider only the balayage $\nu^{\operatorname{bal}^1_{\mathrm{rh}}}$ of genus $1$. In view of this, the first existence results for the balayage $\nu^{\operatorname{bal}^{01}_{\mathrm{rh}}}$ and its order are the very special cases of Theorem 1 in [28]. The finiteness of the upper density of the balayage $\nu^{\operatorname{bal}^1_{\mathrm{rh}}}$ under the additional condition (3.22) is a special case (see [28], assertion 4 in Theorem 3, or [35], Theorem 3.1) in view of Remark 6.
Let us prove that the Lindelöf condition for the difference $\nu -\nu^{\operatorname{bal}^1_{\mathrm{rh}}}$ is met under condition (3.22). This will be a development and strengthening of Theorem 3.2 in [35], where this fact was proved with the additional requirement that the distribution of charges $\nu$ itself satisfies the Lindelöf condition. Without loss of generality we can assume that the distribution of charges $\nu$ is concentrated in $\mathbb{C}_{\mathrm{rh}}$, since the restriction $\nu\lfloor_{\mathbb{C}_{\overline{\mathrm{lh}}}}$ does not change under a balayage from $\mathbb{C}_{\mathrm{rh}}$ and since
$$
\begin{equation*}
\nu-\nu^{\operatorname{bal}_{\mathrm{rh}}^1} = \nu\lfloor_{\mathbb{C}_{\mathrm{rh}}} - (\nu\lfloor_{\mathbb{C}_{\mathrm{rh}}})^{\operatorname{bal}_{\mathrm{rh}}^1}.
\end{equation*}
\notag
$$
If $\nu$ is concentrated in $\mathbb{C}_{\mathrm{rh}}$, then $\operatorname{supp} \nu^{\operatorname{bal}^1_{\mathrm{rh}}}\subset i\mathbb{R}$ and hence by (3.22)
$$
\begin{equation}
\biggl|\int_{(r\overline{\mathbb{D}})\setminus d\overline{\mathbb{D}}}\operatorname{Re} \frac{1}{z}\,d \bigl(\nu-\nu^{\operatorname{bal}^1_{\mathrm{rh}}}\bigr)(z)\biggr|= |\ell_{\nu}^{\operatorname{rh}}(d,r)| \stackrel{(3.22)} {\underset{r\to+\infty}{=}}O(1).
\end{equation}
\tag{3.26}
$$
Thus, the Lindelöf $\mathbb{R}$-condition is satisfied for the difference $\nu-\nu^{\operatorname{bal}^1_{\mathrm{rh}}}$ and it is sufficient to verify the Lindelöf $i\mathbb{R}$-condition for $\nu-\nu^{\operatorname{bal}^1_{\mathrm{rh}}}$.
By (3.25), there exists $C\in \mathbb{R}^+$ such that
$$
\begin{equation}
|\nu|^{\mathrm{rad}}(t)+ \bigl|\nu^{\operatorname{bal}^1_{\mathrm{rh}}}\bigr|^{\mathrm{rad}}(t) \leqslant Ct \quad\text{for all }t\in \mathbb{R}^+.
\end{equation}
\tag{3.27}
$$
For a fixed $r>2$, we represent the distribution of charges $\nu$ as the sum of two distributions of charges
$$
\begin{equation}
\nu :=\nu_{2r}+\nu_{\infty}, \qquad \nu_{2r}:= \nu\lfloor_{2r\overline{\mathbb{D}}}, \qquad \nu_{\infty}:=\nu -\nu_{2r}= \nu\lfloor_{\mathbb{C}\setminus 2r\overline{\mathbb{D}}}.
\end{equation}
\tag{3.28}
$$
By definition of a balayage of genus $1$ (see (3.11)) and from (3.28) we have
$$
\begin{equation}
\begin{aligned} \, &-\int_{(r\overline{\mathbb{D}})\setminus d\overline{\mathbb{D}}} \operatorname{Im} \frac{1}{z} \,d \nu^{\operatorname{bal}^1_{\mathrm{rh}}}(z) \stackrel{(3.11)}{=}\int_{d<|y|\leqslant r} \frac{1}{y}\, d\bigl(\nu^{\operatorname{bal}^1_{\mathrm{rh}}}\bigr)_{i\mathbb{R}}(y) \nonumber \\ &\qquad\stackrel{(3.28)}{=}\int_{d<|y|\leqslant r} \frac{1}{y}\, d \bigl(\nu_{2r}^{\operatorname{bal}^1_{\mathrm{rh}}}\bigr)_{i\mathbb{R}}(y)+ \int_{d<|y|\leqslant r} \frac{1}{y}\, d \bigl(\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}} \bigr)_{i\mathbb{R}}(y). \end{aligned}
\end{equation}
\tag{3.29}
$$
For the first integral from the right-hand side (3.29), twice applying equality (3.16) from Proposition 1 to the continuous odd function $f\colon y\mapsto 1/y$ on the intervals $(d,r]$ and $[-r,-d)$, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\int_{d<|y|\leqslant r} \frac{1}{y}\, d \bigl(\nu_{2r}^{\operatorname{bal}^1_{\mathrm{rh}}}\bigr)_{i\mathbb{R}}(y) \stackrel{(3.16)}{=}\int_{d<|y|\leqslant r} \frac{1}{y}\, d \bigl(\nu_{2r}^{\operatorname{bal}^0_{\mathrm{rh}}}\bigr)_{i\mathbb{R}}(y) \\ &\qquad-\frac{1}{\pi}\int_{\mathbb{C}}\operatorname{Re}\frac{1}{z}\, d \nu(z)\cdot \int_{d<|y|\leqslant r}\frac{1}{y}\,d y =\int_{d<|y|\leqslant r} \frac{1}{y}\,d \bigl(\nu_{2r}^{\operatorname{bal}^0_{\mathrm{rh}}} \bigr)_{i\mathbb{R}}(y), \end{aligned}
\end{equation*}
\notag
$$
where the genus has changed from $1$ to $0$, and a substitution into (3.29) gives
$$
\begin{equation}
\begin{aligned} \, &{-}\int _{(r\overline{\mathbb{D}})\setminus d\overline{\mathbb{D}}} \operatorname{Im} \frac{1}{z} \,d \nu^{\operatorname{bal}^1_{\mathrm{rh}}}(z) \stackrel{(3.10)}{=}\!\!\int_{d<|y|\leqslant r} \frac{1}{y}\, d \bigl(\nu_{2r}^{\operatorname{bal}^0_{\mathrm{rh}}}\bigr)_{i\mathbb{R}}(y) +\int_{d<|y|\leqslant r} \frac{1}{y}\, d \bigl(\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}} \bigr)_{i\mathbb{R}}(y) \nonumber \\ &\quad =\int_{|y|>d} \frac{1}{y}\, d \bigl(\nu_{2r}^{\operatorname{bal}^0_{\mathrm{rh}}}\bigr)_{i\mathbb{R}}(y)+ \int_{|y|>r} -\frac{1}{y}\, d \bigl(\nu_{2r}^{\operatorname{bal}^0_{\mathrm{rh}}}\bigr)_{i\mathbb{R}}(y) + \int_{d<|y|\leqslant r} \frac{1}{y}\, d \bigl(\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}} \bigr)_{i\mathbb{R}}(y). \end{aligned}
\end{equation}
\tag{3.30}
$$
For the second integral on the right of (3.30) we obtain the inequalities
$$
\begin{equation}
\biggl|\int_{|y|>r} - \frac{1}{y}\,d \bigl(\nu_{2r}^{\operatorname{bal}^0_{\mathrm{rh}}} \bigr)_{i\mathbb{R}}(y)\bigr| \stackrel{(3.28)}{\leqslant} \frac{1}{r}\bigl|\nu_{2r}^{\operatorname{bal}^0_{\mathrm{rh}}} \bigr|^{\mathrm{rad}}(2r) \stackrel{(3.9)}{\leqslant} \frac{1}{r}|\nu_{2r}|^{\mathrm{rad}}(2r) \stackrel{(3.27)}{\leqslant} 2C.
\end{equation}
\tag{3.31}
$$
For the last integral on the right of (3.30), we have
$$
\begin{equation}
\begin{aligned} \, &\biggl|\int_{d<|y|\leqslant r}\frac{1}{y}\ d (\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}})_{i\mathbb{R}}(y) \bigr|\leqslant \int_d^r \frac{1}{t}\, d \bigl|\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}} \bigr|^{\mathrm{rad}}(t) \nonumber \\ &\qquad\leqslant \frac{\bigl|\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}} \bigr|^{\mathrm{rad}}(r)}{r}+ \int_d^r\frac{\bigl|\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}} \bigr|^{\mathrm{rad}}(t)}{t^2} \,d t \stackrel{(3.27)}{\leqslant} C+ \int_d^r\frac{\bigl|\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}} \bigr|^{\mathrm{rad}}(t)}{t^2} \,d t. \end{aligned}
\end{equation}
\tag{3.32}
$$
Using definition (3.10) of the harmonic charge $\Omega_{\mathrm{rh}}$ and Lemma 3.1 in [35], we have
$$
\begin{equation}
|\Omega_{\mathrm{rh}}(z,i[-t,t])|\leqslant 2\frac{t^2}{|z|^2} \quad \text{when } 2t\leqslant |z|.
\end{equation}
\tag{3.33}
$$
Since $\operatorname{supp} \nu_{\infty} \stackrel{(3.28)}{\subset} \mathbb{C}\setminus 2r\mathbb{D}$, we have from this inequality
$$
\begin{equation*}
\begin{aligned} \, \bigl|\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}}\bigr|^{\mathrm{rad}}(t) &\stackrel{(3.11)}{\leqslant} \int_{|z|\geqslant 2r} |\Omega_{\mathrm{rh}}(z, i[-t,t])|\,d |\nu_{\infty}| (z) \stackrel{(3.33)}{\leqslant} 2t^2\int_{2r}^{\infty} \frac{d |\nu_{\infty}|^{\mathrm{rad}}(s)}{s^2} \\ &\ \ \leqslant 4t^2\int_{2r}^{\infty} \frac{|\nu|^{\mathrm{rad}}(s)}{s^3}\,d s \stackrel{(3.27)}{\leqslant} 4t^2\int_{2r}^{\infty} \frac{Cs}{s^3}\,d s= 2C\frac{t^2}{r} \quad\text{when }t\leqslant r. \end{aligned}
\end{equation*}
\notag
$$
Applying this to the integral on the right of (3.32), this gives
$$
\begin{equation*}
\biggl|\int_{d<|y|\leqslant r} \frac{1}{y}\,d (\nu_{\infty}^{\operatorname{bal}^1_{\mathrm{rh}}})_{i\mathbb{R}}(y)\biggr| \leqslant C+\int_d^r\frac{2C{t^2}/{r}}{t^2} \,d t\leqslant 3C.
\end{equation*}
\notag
$$
An application of this estimate with (3.31) to (3.30) shows that
$$
\begin{equation}
\biggl|\int_{|y|>d} \frac{1}{y}\,d \bigl(\nu_{2r}^{\operatorname{bal}^0_{\mathrm{rh}}}\bigr)_{i\mathbb{R}}(y) + \int_{(r\overline{\mathbb{D}})\setminus d\overline{\mathbb{D}}} \operatorname{Im} \frac{1}{z} \,d \nu^{\operatorname{bal}^1_{\mathrm{rh}}} (z)\biggr|\leqslant 5C \quad\text{for all }r>2.
\end{equation}
\tag{3.34}
$$
For the distribution of charges $\nu_{2r}$ concentrated in $\mathbb{C}_{\mathrm{rh}}$ and with compact support in $\mathbb{C}_{\overline{\mathrm{rh}}}$, the first integral
$$
\begin{equation}
\int_{|y|>d} \frac{1}{y}\,d \bigl(\nu_{2r}^{\operatorname{bal}^0_{\mathrm{rh}}}\bigr)_{i\mathbb{R}}(y)
\end{equation}
\tag{3.35}
$$
under the modulus sign in the left-hand side of (3.34) is equal to (see Theorem 1.2 in [48], Theorem 6 in [9], and formula 3.14 in [35]) the integral over the distribution of charges $\nu_{2r}$ of the Poisson integral in $\mathbb{C}_{\mathrm{rh}}$ of the function
$$
\begin{equation}
iy\underset{y\in \mathbb{R}}{\longmapsto} \begin{cases} \dfrac1{y} &\text{if } |y|>d, \\ 0 &\text{if }|y|\leqslant d, \end{cases}
\end{equation}
\tag{3.36}
$$
on $i\mathbb{R}$, that is, of the harmonic continuation of function (3.36) to $\mathbb{C}_{\mathrm{rh}}$. This continuation can be written explicitly (see formula (3.15) in [35]), and at points $z\in \mathbb{C}_{\mathrm{rh}}$ it assumes the value
$$
\begin{equation*}
\biggl(\frac{1}{\pi}\operatorname{Re} \frac{1}{z} \ln\biggl|\frac{z-id}{z+id}\biggr| +\operatorname{Im} \frac{1}{z}\, \omega_{\mathrm{rh}}(z, i[-d,d]) \biggr) -\operatorname{Im} \frac{1}{z}.
\end{equation*}
\notag
$$
Hence integral (3.35) is equal to
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mathbb{C}}\biggl(\frac{1}{\pi}\operatorname{Re} \frac{1}{z} \ln\biggl|\frac{z-id}{z+id}\biggr|+\operatorname{Im} \frac{1}{z}\, \omega_{\mathrm{rh}} (z, i[-d,d])\biggr) \,d \nu_{2r}(z)-\int_{\mathbb{C}} \operatorname{Im} \frac{1}{z}\,d \nu_{2r}(z) \\ &\stackrel{(3.28),(3.25)}{=} \int_{2d<|z|\leqslant 2r} \biggl(\frac{1}{\pi}\operatorname{Re} \frac{1}{z} \ln\biggl|\frac{z-id}{z+id}\biggr| +\operatorname{Im} \frac{1}{z}\, \omega_{\mathrm{rh}} (z, i[-d,d])\biggr) \,d \nu(z) \\ &\qquad\qquad-\int_{r<|z|\leqslant 2r}\operatorname{Im} \frac{1}{z}\,d \nu(z) - \int_{r\mathbb{D}\setminus d\overline{\mathbb{D}}}\operatorname{Im} \frac{1}{z}\,d \nu(z). \end{aligned}
\end{equation*}
\notag
$$
Substituting the right-hand side into (3.34) gives
$$
\begin{equation}
\begin{aligned} \, &\biggl|-\int_{(r\overline{\mathbb{D}})\setminus d\overline{\mathbb{D}}} \operatorname{Im} \frac{1}{z}\,d \nu(z)+\int _{(r\overline{\mathbb{D}}) \setminus d\overline{\mathbb{D}}}\operatorname{Im} \frac{1}{z} \, d \nu^{\operatorname{bal}^1_{\mathrm{rh}}}(z)\biggr| \nonumber \\ &\ \leqslant 5C+\biggl|-\int_{r<|z|\leqslant 2r}\operatorname{Im} \frac{1}{z}\,d \nu(z)\biggr| \nonumber \\ &\ \qquad+ \biggl|\int_{2d<|z|\leqslant 2r}\biggl(\frac{1}{\pi}\operatorname{Re} \frac{1}{z} \ln\biggl|\frac{z-id}{z+id}\biggr|+ \operatorname{Im} \frac{1}{z}\,\omega_{\mathrm{rh}} (z,i[-d,d])\biggr) \, d \nu(z)\biggr| \nonumber \\ &\ \leqslant 5C+\frac{1}{r}|\nu|^{\mathrm{rad}}(2r) \nonumber \\ &\ \qquad+\int _{2d<|z|\leqslant 2r} \biggl(\frac{1}{\pi} \biggl|\ln\biggl|\frac{z-id}{z+id}\biggr|\biggr|+ \omega_{\mathrm{rh}} (z, i[-d,d])\biggr)\frac{d |\nu|(z)}{|z|} \quad\text{for all }r>2. \end{aligned}
\end{equation}
\tag{3.37}
$$
For $|z|\geqslant 2d$, we have $|z\pm id|\geqslant |z|/2$. Hence from
$$
\begin{equation*}
- \ln \biggl(1+\frac{2d}{|z-id|}\biggr)\leqslant \ln\biggl|\frac{z-id}{z+id}\biggr| \leqslant \ln \biggl(1+\frac{2d}{|z+id|}\biggr)
\end{equation*}
\notag
$$
we find that
$$
\begin{equation}
\biggl|\ln\biggl|\frac{z-id}{z+id}\biggr|\biggr|\leqslant \ln \biggl(1+ \frac{4d}{|z|}\biggr)\leqslant \frac{4d}{|z|}\leqslant \frac{4}{|z|} \quad\text{for all }|z|\geqslant 2d.
\end{equation}
\tag{3.38}
$$
For $|z|\geqslant 2d$, we have $|z|- d\geqslant |z|/2$. By Proposition 3.1, formula (3.10) in [9],
$$
\begin{equation}
\omega_{\mathrm{rh}}(z, i[-d,d])\leqslant \frac{1}{\pi} \frac{2d\operatorname{Re} z}{|z|^2-d^2}\leqslant \frac{1}{\pi} \frac{4d\operatorname{Re} z}{|z|^2}\leqslant \frac{1}{\pi}\frac{4}{|z|} \quad\text{for all }|z|\geqslant 2d.
\end{equation}
\tag{3.39}
$$
Using (3.38), (3.39), from inequalities (3.37) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\biggl|\int_{(r\overline{\mathbb{D}})\setminus d\overline{\mathbb{D}}} \operatorname{Im} \frac{1}{z}\,d \bigl(\nu- \nu^{\operatorname{bal}^1_{\mathrm{rh}}}\bigr)(z)\biggr| \\ &\qquad\stackrel{(3.27)}{\leqslant} 7C +\int _{2d<|z|\leqslant 2r} \biggl(\frac{1}{\pi}\biggl|\ln\biggl|\frac{z-id}{z+id}\biggr|\biggr|+ \omega_{\mathrm{rh}} (z, i[-d,d])\biggr)\frac{d \nu(z)}{|z|} \\ &\stackrel{(3.38),(3.39)}{\leqslant} 7C +\frac{8}{\pi} \int _{2d<|z|\leqslant 2r} \frac{d \nu(z)}{|z|^2} \leqslant 7C +3 \int _{2d}^{+\infty} \frac{d |\nu|^{\mathrm{rad}}(t)}{t^2} \\ &\stackrel{(3.27)}{\leqslant} 7C +6 \int _{2d}^{+\infty} \frac{|\nu|^{\mathrm{rad}}(t)}{t^3}\,d t \stackrel{(3.27)}{\leqslant} 7C + 6 C\int _{2d}^{+\infty} \frac{\,d t}{t^2} \\ &\ \, = 7C +3\frac{C}{d}\quad\text{for all }r\geqslant 2. \end{aligned}
\end{equation*}
\notag
$$
Thus, the difference $\nu-\nu^{\operatorname{bal}^1_{\mathrm{rh}}}$ satisfies the Lindelöf condition.
Relation (3.24) under conditions (3.22) and (3.23) is a special case of a combination of Corollary 4.2,(ii) in [9] for a balayage of genus $0$ and Corollary 3.1,(ii), formula (3.24) in [28] for a balayage of genus $1$, which is implicitly reflected in [35], Theorem 3.3, formula (3.18). This proves Proposition 2. 3.2. Balayage of differences of subharmonic functionsLet $\mathcal U=u-v$ be the difference of subharmonic functions $u$ and $v$ on $\mathbb{C}$, that is, $\mathcal U$ a $\delta$-subharmonic function. We write ${\mathcal U}\not\equiv \pm\infty$ if $u\not\equiv -\infty$ and $v \not\equiv -\infty$. The values of such a function $\mathcal{U}\not\equiv \pm\infty$ are defined at all points at which one of the functions $u$ or $v$ takes a value from $\mathbb{R}$, that is, outside some polar set. Its Riesz distribution of charges
$$
\begin{equation}
\varDelta_\mathcal{U}:= \frac{1}{2\pi}\Delta \mathcal{U}\stackrel{(1.12)}{:=} \frac{1}{2\pi}\Delta u-\frac{1}{2\pi}\Delta v\stackrel{(1.12)}{=} \varDelta_u-\varDelta_v
\end{equation}
\tag{3.40}
$$
is the difference between the Riesz distributions of masses of the functions $u$ and $v$. By Definition 4.1 in [28], a $\delta$-subharmonic balayage of a $\delta$-subharmonic function $\mathcal{U}\not\equiv \pm\infty$ from $\mathbb{C}_{\mathrm{rh}}$ onto $\mathbb{C}_{\overline{\mathrm{lh}}}$ is a $\delta$-subharmonic function on $\mathbb{C}$ which is harmonic on $\mathbb{C}_{\mathrm{rh}}$ and equal to $\mathcal{U}$ on the closed left half-plane $\mathbb{C}_{\overline{\mathrm{lh}}}$ outside some polar set. Proposition 3 (a special case of Theorems 6, 7 in [28]). Let a $\delta$-subharmonic function $\mathcal{U}\not\equiv \pm \infty$ with Riesz distribution of charges (3.40) of finite upper density be the difference of subharmonic functions of order $\leqslant 1$. Then there exists a $\delta$-subharmonic balayage $\mathcal U^{\operatorname{Bal}_{\mathrm{rh}}} \not\equiv \pm\infty$ from $\mathbb{C}_{\mathrm{rh}}$ onto $\mathbb{C}_{\overline{\mathrm{lh}}}$ with Riesz distribution of charges Proof. By Theorem 6 in [28], for any $\delta$-subharmonic function $\mathcal U$ with Riesz distribution of charges of finite type, there exists a balayage $\mathcal{U}^{\operatorname{Bal}_{\mathrm{rh}}}$ with Riesz distribution of charges (3.41), representable in the form
$$
\begin{equation}
\mathcal{U}^{\operatorname{Bal}_{\mathrm{rh}}}=v_+-u_-+H, \qquad v_+\not\equiv -\infty, \quad u_-\not\equiv -\infty,
\end{equation}
\tag{3.43}
$$
where $v_+$ and $u_-$ are subharmonic functions of order $\leqslant 1$, and $H$ is a harmonic function on $\mathbb{C}$. Moreover, if $\mathcal U$ is representable as the difference of subharmonic functions of order $\leqslant 1$, then, by the final part of Theorem 6 in [28], we can choose as $H|$ a harmonic polynomial of degree $\deg H\leqslant 1$. Thus, $u_+:=v_++H\not\equiv -\infty$ is a subharmonic function of order $\leqslant 1$, and so (3.42) follows from (3.43). The possibility of replacing the right-hand side in (3.41) by $\varDelta_\mathcal{U}^{\operatorname{bal}^{1}_{\mathrm{rh}}}$ follows from (3.14) in Remark 7. This proves Proposition 3. § 4. Two constructions with distributions of charges and their balayage associated with logarithmic functions of intervalsIf the function $z \mapsto \operatorname{Re} (1/z)$ is summable over the total variation $|\nu|$ of the distribution of charges $\nu $ in the right neighbourhood of zero, that is,
$$
\begin{equation}
\int_{\mathbb{D}\cap \mathbb{C}_{\mathrm{rh}}}\operatorname{Re} \frac{1}{z}\,d |\nu|(z)<+\infty \quad \Longleftrightarrow\quad \lim_{0<r\to 0} \ell_{|\nu|}^{\operatorname{rh}} (r,1) <+\infty,
\end{equation}
\tag{4.1}
$$
or the function $z \mapsto \operatorname{Re}(-1/z)$ is summable over $|\nu|$ in the left neighbourhood of zero, that is,
$$
\begin{equation}
\int_{\mathbb{D}\cap \mathbb{C}_{\mathrm{lh}}}\operatorname{Re} \frac{-1}{z}\,d |\nu|(z)<+\infty \quad \Longleftrightarrow\quad \lim_{0<r\to 0} \ell_{|\nu|}^{\operatorname{lh}} (r,1) <+\infty,
\end{equation}
\tag{4.2}
$$
then the right and left characteristic logarithms of $\nu$
$$
\begin{equation}
\begin{aligned} \, \ell_{\nu}^{\operatorname{rh}}(R)&:=\ell_{\nu}^{\operatorname{rh}}(0,R) \stackrel{(4.1)}{:=} \lim_{0<r\to 0}\ell_{\nu}^{\operatorname{rh}}(r,R), \\ \ell_{\nu}^{\operatorname{lh}}(R)&:=\ell_{\nu}^{\operatorname{lh}}(0,R) \stackrel{(4.2)}{:=}\lim_{0<r\to 0} \ell_{\nu}^{\operatorname{lh}}(r,R) \end{aligned}
\end{equation}
\tag{4.3}
$$
are defined by continuity, for $0<r\to 0$ (these logarithms were defined in [7], § 2.2, and [3], Ch. 22, only for positive distributions of points. Besides, the two-sided characteristic logarithm of the measure $\mu$ can be defined by
$$
\begin{equation*}
\ell_{\mu}(R):=\ell_{\mu}(0,R):=\lim_{0<r\to 0}\ell_{\mu}(r,R).
\end{equation*}
\notag
$$
Related versions of the following proposition for distributions of points can be found in [7], Lemma 3.1, [3], Lemma 22.2, [23], Lemma 1.1, and [35], Lemma 1. Proposition 4. For any distribution of charges $\eta$ on $\mathbb{C}$, $0\notin \operatorname{supp} \eta$, there is a distribution of masses $\alpha$ such that $\operatorname{supp} \alpha\subset \mathbb{R}^+\setminus 0$ and in addition, this distribution of masses $\alpha$ has finite upper density if so is $\eta$. Proof. The right-hand side in (4.4) cannot be equal to $-\infty$. If the right-hand side in (4.4) is $+\infty$, then we set $\alpha:=0$, and in this case $\leqslant +\infty$. In other cases, the function
$$
\begin{equation}
a\colon t \stackrel{(2.1)}{\underset{t\in \mathbb{R}^+ }{\longmapsto}} -\sup_{s\geqslant t} \ell_{\eta}^{\operatorname{rh}}(s) = \inf_{s\geqslant t} \bigl(-\ell_{\eta}^{\operatorname{rh}}(s)\bigr),
\end{equation}
\tag{4.5}
$$
which is increasing by construction, uniquely defines the distribution of masses $\alpha$ via its increasing distribution function
$$
\begin{equation}
\alpha_{\mathbb{R}}\colon x\underset{x\in \mathbb{R}^+ }{\longmapsto} \int_0^x t \,d a(t)=xa(x)-\int_0^xa(t)\,d t.
\end{equation}
\tag{4.6}
$$
By construction (4.5) and the condition $ 0\notin \operatorname{supp} \eta$, the function $a$ is constant on some interval $[0,r_0)\neq \varnothing$, whence, for the distribution function (4.6), we have $\alpha_{\mathbb{R}}\equiv 0$ on $[0,r_0)$ and $\operatorname{supp} \alpha\subset \mathbb{R}^+\setminus 0$. By construction (4.5), we also have
$$
\begin{equation}
a(t) \stackrel{(4.5)}{=}\inf_{s\geqslant t} \bigl( -\ell_{\eta}^{\operatorname{rh}}(s)\bigr)\leqslant - \ell_{\eta}^{\operatorname{rh}}(t) \quad\text{for all } t\in \mathbb{R}^+,\text{ whence }a(0)\leqslant 0,
\end{equation}
\tag{4.7}
$$
$$
\begin{equation}
a(t) \stackrel{(4.5)}{=} -\sup_{s\geqslant t} \ell_{\eta}^{\operatorname{rh}}(s) \geqslant -\sup_{0\leqslant r<R<+\infty} \ell_{\eta}^{\operatorname{rh}}(r,R) \quad\text{for all } t\in \mathbb{R}^+.
\end{equation}
\tag{4.8}
$$
In addition, by definition (4.5), for all $t\in \mathbb{R}^+$ we have
$$
\begin{equation*}
\ell_{\alpha}^{\operatorname{rh}} (t)\stackrel{(4.3)}{=} \int_0^t\frac{1}{x}\,d \alpha_{\mathbb{R}}(x)\stackrel{(4.6)}{:=} \int_0^t \,d a(t)=a(t)-a(0),
\end{equation*}
\notag
$$
and hence, for each $t\in \mathbb{R}^+$,
$$
\begin{equation}
\ell_{\eta+\alpha}^{\operatorname{rh}}(t) =\ell_{\eta}^{\operatorname{rh}}(t)+a(t)-a(0)\stackrel{(4.7)}{\leqslant} -a(0)\stackrel{(4.8)}{\leqslant} \sup_{0\leqslant r<R<+\infty} \ell_{\eta}^{\operatorname{rh}}(r,R),
\end{equation}
\tag{4.9}
$$
and
$$
\begin{equation*}
\begin{aligned} \, \ell_{\eta+\alpha}^{\operatorname{rh}}(t) &\stackrel{(4.9)}{=} \ell_{\eta}^{\operatorname{rh}}(t)+a(t)-a(0) \stackrel{(4.5)}{=} \inf_{s\geqslant t}\bigl(\ell_{\eta}^{\operatorname{rh}}(t) -\ell_{\eta}^{\operatorname{rh}}(s)\bigr)-a(0) \\ &\ =-\sup_{s\geqslant t} \bigl(\ell_{\eta}^{\operatorname{rh}}(s)- \ell_{\eta}^{\operatorname{rh}}(t) \bigl)-a(0)\stackrel{(4.3)}{=} -\sup_{s\geqslant t}\bigl(\ell_{\eta}^{\operatorname{rh}}(t,s) \bigl)-a(0) \\ &\stackrel{(4.7)}{\geqslant} -\sup_{0\leqslant r<R<+\infty} \ell_{\eta}^{\operatorname{rh}}(r,R)\quad\text{for any }t\in \mathbb{R}^+. \end{aligned}
\end{equation*}
\notag
$$
Hence, using (4.9), we find that
$$
\begin{equation*}
\sup_{t\geqslant 0} |\ell_{\eta+\alpha}^{\operatorname{rh}}(t)|\leqslant \sup_{0\leqslant r<R<+\infty} \ell_{\eta}^{\operatorname{rh}}(r,R),
\end{equation*}
\notag
$$
which entails (4.4), inasmuch as
$$
\begin{equation*}
\sup_{0<r<R<+\infty} |\ell_{\eta+\alpha}^{\operatorname{rh}}(r,R)|\leqslant \sup_{R\in \mathbb{R}^+} |\ell_{\eta+\alpha}^{\operatorname{rh}}(R)r|+ \sup_{r\in \mathbb{R}^+} |\ell_{\eta+\alpha}^{\operatorname{rh}}(r)|\leqslant 2 \sup_{0\leqslant r<R<+\infty} \ell_{\eta}^{\operatorname{rh}}(r,R).
\end{equation*}
\notag
$$
If $\eta$ is the distribution of charges of finite upper density, then
$$
\begin{equation}
|\ell_{\eta}^{\operatorname{rh}}(r,2r)|\leqslant \int_r^{2r} \frac{1}{t}\,d |\eta|^{\mathrm{rad}}(t)\leqslant \frac{1}{r}|\eta|^{\mathrm{rad}}(2r)\underset{r\to +\infty}{=}O(1).
\end{equation}
\tag{4.10}
$$
Hence by(4.4) we have
$$
\begin{equation*}
\begin{aligned} \, \frac{1}{2r}\bigl(\alpha^{\mathrm{rad}}(2r)-\alpha^{\mathrm{rad}}(r)\bigr) &\leqslant \int_r^{2r} \frac{1}{t}\,d \alpha^{\mathrm{rad}}(t) =\ell_{\alpha}^{\operatorname{rh}}(r,2r) \leqslant |\ell_{\eta+\alpha}^{\operatorname{rh}}(r,2r)| + |\ell_{\eta}^{\operatorname{rh}}(r,2r)| \\ &\!\!\!\stackrel{(4.4)}{\leqslant} 2\sup_{0\leqslant r<R<+\infty} \ell_{\eta}^{\operatorname{rh}}(r,R)+|\ell_{\eta}^{\operatorname{rh}}(r,2r)| \underset{r\to +\infty}{\stackrel{(4.10)}{=}}O(1). \end{aligned}
\end{equation*}
\notag
$$
This means that the distribution of masses $\alpha$ has finite upper density. Proposition 4 is proved. Proposition 5. Let $a \in (0,1)$. Suppose that the support of distributions of masses $\nu$ and $\mu$ of finite type are contained in the angle $\{z\in \mathbb{C}\mid \operatorname{Re} z > a|z|\}$, and also Then there exists a distribution of masses $\alpha$ of finite upper density with support on $\mathbb{R}^+\setminus 0$ and with balayage $(\nu+\alpha-\mu)^{\operatorname{bal}^1_{\mathrm{rh}}}$ of genus $1$ from $\mathbb{C}_{\mathrm{rh}}$ onto $ \mathbb{C}_{\overline{\mathrm{lh}}}$ of finite upper density with support on $i\mathbb{R}$ and also there are a distribution of masses $\beta$ with support on $i\mathbb{R}$ and the number $c\in \mathbb{R}^+$ such that is the linear Lebesgue measure on the imaginary axis multiplied by $c$, and the charge distribution $\nu+\alpha+\beta-\mu$ satisfies the Lindelöf condition. Proof. For the charge distribution $\eta:=\nu-\mu$, from condition (4.11) and Proposition 4 it follows that there exists a mass distribution $\alpha$ of finite upper density such that $\operatorname{supp} \alpha \subset \mathbb{R}^+\setminus 0$ and
$$
\begin{equation*}
\sup_{0\leqslant r<R<+\infty} |\ell_{\nu+\alpha-\mu}^{\operatorname{rh}}(r,R)| = \sup_{0\leqslant r<R<+\infty} |\ell_{\eta +\alpha}^{\operatorname{rh}}(r,R)| \stackrel{(4.4)}{<}+\infty.
\end{equation*}
\notag
$$
This means that the supports of $\nu$ and $\mu$ lie in $\{z\in \mathbb{C}\mid \operatorname{Re} z> a|z|\}$, and, in addition, that conditions (3.22) and (3.23) of Proposition 2 are satisfied with $\nu+\alpha-\mu$ in place of $\nu$. By Proposition 2, there exists a balayage $(\nu+\alpha-\mu)^{\operatorname{bal}^1_{\mathrm{rh}}}$ of finite upper density concentrated in this case exclusively on $i\mathbb{R}$ such that the difference $(\nu+\alpha-\mu)-(\nu+\alpha-\mu)^{\operatorname{bal}^1_{\mathrm{rh}}}$ satisfies the Lindelöf condition and
$$
\begin{equation}
\sup_{y\in \mathbb{R}}\sup_{t\in (0,1]}\frac{\bigl|(\nu+\alpha- \mu)^{\operatorname{bal}^1_{\mathrm{rh}}}\bigr|_{iy}(t)}{t} \stackrel{(3.24)}{<}+\infty.
\end{equation}
\tag{4.13}
$$
We set
$$
\begin{equation}
(\nu +\alpha -\mu)^{\operatorname{bal}^1_{\mathrm{rh}}}=: \vartheta= \vartheta^+-\vartheta^-,
\end{equation}
\tag{4.14}
$$
where $\vartheta^+$ and $\vartheta^-$ are the upper and lower variations of the distribution of charges $\vartheta$. By construction, the distributions of masses $\vartheta^\pm$ are of finite upper density and have supports in $i\mathbb{R}$. In view of (4.13), for some $c\in \mathbb{R}^+$, we have
$$
\begin{equation}
\vartheta^{\pm}_{iy}(t)\leqslant 2ct \quad\text{for all } y\in \mathbb{R}\text{ and }t\in (0,1].
\end{equation}
\tag{4.15}
$$
Hence, for the linear Lebesgue measure $\mathfrak m_1\lfloor_{i\mathbb{R}}$ on $i\mathbb{R}$, the difference $c\mathfrak m_1\lfloor_{i\mathbb{R}}-\vartheta^+$ is the distribution of masses of finite upper density with support on $i\mathbb{R}$. From (4.14) we have
$$
\begin{equation}
(\nu +\alpha -\mu)^{\operatorname{bal}^1_{\mathrm{rh}}} + \underset{\beta}{\underbrace{\vartheta^-+(c\mathfrak m_1\lfloor_{i\mathbb{R}}- \vartheta^+)}} \stackrel{(4.14)}{=}\vartheta^++ c\mathfrak m_1\lfloor_{i\mathbb{R}}-\vartheta^+= c\mathfrak m_1\lfloor_{i\mathbb{R}}.
\end{equation}
\tag{4.16}
$$
For this distribution mass $\beta:=\vartheta^-+ (c\mathfrak m_1\lfloor_{i\mathbb{R}}-\vartheta^+)$ of finite upper density, equality (4.16) means that the second equality in (4.12) holds. In addition, $\operatorname{supp} \beta\subset i\mathbb{R}$. Hence by definition (3.11) of a balayage of genus $1$ from $\mathbb{C}_{\mathrm{rh}}$, we have $\beta^{\operatorname{bal}^1_{\mathrm{rh}}}=\beta$, which gives the first equality in (4.12). Finally, $(\nu+\alpha-\mu)-(\nu+\alpha-\mu)^{\operatorname{bal}^1_{\mathrm{rh}}}$ satisfies the Lindelöf condition, as noted before (4.13). From the explicit form of the distribution of masses $c\mathfrak m_1|_{i\mathbb{R}}$ from the right-hand side of (4.12) (which, clearly, satisfies the Lindelöf condition) and from the second equality in (4.12) it follows that the distribution of charges $(\nu+\alpha -\mu)^{\operatorname{bal}^1_{\mathrm{rh}}}+\beta $ satisfies the Lindelöf condition. Therefore, their sum $(\nu+\alpha-\mu)+\beta=\nu+\alpha+\beta -\mu$, also satisfies the Lindelöf condition. This proves Proposition 5. § 5. Balayage onto a vertical strip5.1. Shifts and two-sided balayage of distribution of chargesThe reflection symmetry $z\underset{z\in \mathbb{C}}{\longmapsto} -\overline z$ with respect to the imaginary axis allows us to reformulate all the results on the balayage of genus $1$ from $\mathbb{C}_{\mathrm{rh}}$ onto $\mathbb{C}_{\overline{\mathrm{lh}}}$ for a balayage from the left half-plane $ \mathbb{C}_{\mathrm{lh}}$ onto $\mathbb{C}_{\overline{\mathrm{rh}}}$ with replacement, where necessary, of the right logarithmic function of intervals (2.1) by the left logarithmic function of intervals (2.2), redenoting the superscript $^{\operatorname{bal}^1_{\mathrm{rh}}}$ by $^{\operatorname{bal}^1_{\mathrm{lh}}}$ for balayage from the left half-plane $\mathbb{C}_{\mathrm{lh}}$. Let $\nu$ be a distribution of charges and $w\in \mathbb{C}$. By $\nu_{\vec{w}}$ we denote the $w$-shift of charge distribution $\nu$ defined by
$$
\begin{equation}
\nu_{\vec{w}}(K):=\nu(K-w) \quad\text{on compact sets }K\subset \mathbb{C}.
\end{equation}
\tag{5.1}
$$
Proposition 6. Let $\nu$ be a distribution of charges of finite upper density. Then, for any $w\in \mathbb{C}$ and $r_0\in \mathbb{R}^+\setminus 0$, We omit the proof of Proposition 6, which easily follows from the definitions of $\ell^{\operatorname{rh}}$ and $\ell^{\operatorname{lh}}$ (see (2.1)–(2.2)) and the Lindelöf conditions (3.18)–(3.21). We will describe the balayage of genus $01$ of the distribution of charges $\nu$ onto the closed vertical strip $\overline{\operatorname{str}}_b$ of width $2b\geqslant 0$ from (1.5) in five steps [b1]–[b5]. Each following step is applied to the distribution of charges obtained in the previous step: [b1] the $(-b)$-shift $\nu_{\vec{-b}}$ of distribution of charges $\nu$; [b2] the balayage $\nu_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}}$ of genus $01$ from the right half-plane $\mathbb{C}_{\mathrm{rh}}$ onto $ \mathbb{C}_{\overline{\mathrm{lh}}}$; [b3] the $2b$-shift $\Bigl(\nu_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}} \Bigr)_{\vec{2b}}$ of the distribution of charges $\nu_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}}$; [b4] the balayage $\Bigl(\nu_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}} \Bigr)_{\vec{2b}}^{{\operatorname{bal}^{01}_{\mathrm{lh}}}}$ of genus $01$ from the left half-plane $ \mathbb{C}_{\mathrm{lh}}$ onto $\mathbb{C}_{\overline{\mathrm{rh}}}$; [b5] the $(-b)$-shift $\bigl(\bigl(\nu_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}} \bigr)_{\vec{2b}}^{{\operatorname{bal}^{01}_{\mathrm{lh}}}} \bigr)_{\vec{-b}}$ of distribution of charges $\bigl(\nu_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}} \bigr)_{\vec{2b}}^{{\operatorname{bal}^{01}_{\mathrm{lh}}}}$. For brevity, the distribution of charges obtained in step [b5] will be denoted by
$$
\begin{equation}
\nu^{\operatorname{Bal}^{01}_b}:= \biggl(\Bigl(\nu_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}} \Bigr)_{\vec{2b}}^{{\operatorname{bal}^{01}_{\mathrm{lh}}}}\biggr)_{\vec{-b}}
\end{equation}
\tag{5.3}
$$
and called the balayage of genus $01$ onto $\overline{\operatorname{str}}_b$ of distribution of charges $\nu$ provided that steps [b2] and [b4] can be realized. By Proposition 2, a sufficient condition for realization of steps [b2] and [b4] is that the distribution of charges $\nu$ belong to the convergence class for order $p\stackrel{(3.17)}{=}2$. Remark 8. By Remark7, if $\pm b\notin \operatorname{supp} \nu$ or, more generally, if Proposition 7. Let $\mu$ be a distribution of masses of finite upper density satisfying the Lindelöf condition and let $\nu$ be a distribution of masses such that Proof. Let us verify that $\nu$ is of finite upper density. Indeed, from (5.6), since $\operatorname{type}[\mu]<+\infty$, we have, for some $C\in \mathbb{R}$,
$$
\begin{equation*}
\begin{aligned} \, \ell_{\nu}(r,2r) &\leqslant \ell_{\mu}(r,2r)+ C\leqslant \int_r^{2r} \frac{1}{|z|}\,d \mu(z)+C \\ &\leqslant \frac{1}{r}\bigl(\mu^{\mathrm{rad}}(2r)-\mu^{\mathrm{rad}}(r)\bigr)+ C\leqslant 2\operatorname{type}[\mu]+1+C \end{aligned}
\end{equation*}
\notag
$$
for sufficiently large $r\in \mathbb{R}^+$. By (5.7), we also have the lower estimate
$$
\begin{equation}
\begin{aligned} \, \ell_{\nu}(r,2r) &\stackrel{(2.3)}{\geqslant} \ell_\nu^{\operatorname{rh}}(r, 2r) \stackrel{(2.1)}{=} \int_{r}^{2r}\frac{\operatorname{Re} z}{|z|^2} \, d \nu\lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}(z) \nonumber \\ &\stackrel{(5.7)}{\geqslant} \int_{r}^{2r}\frac{a|z|}{|z|^2} \,d \nu \lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}(z) \geqslant \frac{a}{2r} \bigl(\nu\lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}^{\mathrm{rad}}(2r) - \nu\lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}^{\mathrm{rad}}(r)\bigr), \end{aligned}
\end{equation}
\tag{5.9}
$$
whence, by the previous upper estimate, for sufficiently large $r\in\mathbb{R}^+$. Hence the distribution of masses $\nu\lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}$ is of finite upper density. A similar analysis shows that $\nu\lfloor_{\mathbb{C}_{\overline{\mathrm{lh}}}}$ is of finite upper density. Hence so is $\nu$.
By the assumption, the distribution of masses $\mu$ satisfies the Lindelöf condition (3.21), and hence $\mu$ satisfies the $\mathbb{R}$-Lindelöf condition (3.18)–(3.19), and
$$
\begin{equation*}
\begin{aligned} \, &\sup_{1\leqslant r<R<+\infty} |\ell^{\operatorname{rh}}_{\mu}(r,R)- \ell^{\operatorname{lh}}_{\mu}(r,R)| \stackrel{(2.1),(2.2)}{=}\sup_{1\leqslant r<R<+\infty} \biggl| \int_{r<|z|\leqslant R}\operatorname{Re}\frac{1}{z}\,d \mu(z)\biggr| \\ &\qquad\leqslant \sup_{r\geqslant 1} |\ell^{\operatorname{rh}}_{\mu}(1,r)- \ell^{\operatorname{lh}}_{\mu}(1,r)| + \sup_{R\geqslant 1} |\ell^{\operatorname{rh}}_{\mu}(1,R) - \ell^{\operatorname{lh}}_{\mu}(1,R)|\stackrel{(3.19)}{<}+\infty, \end{aligned}
\end{equation*}
\notag
$$
whence, by definition (2.3) of the logarithmic submeasure of intervals for $\mu$,
$$
\begin{equation}
\sup_{1\leqslant r<R<+\infty} |\ell_{\mu}(r,R)- \ell^{\operatorname{rh}}_{\mu}(r,R)|+ \sup_{1\leqslant r<R<+\infty} |\ell_{\mu}(r,R) - \ell^{\operatorname{lh}}_{\mu}(r,R)|<+\infty.
\end{equation}
\tag{5.10}
$$
From condition (5.6) and from definition (2.3) of the logarithmic submeasure of intervals for $\nu$, we obtain
$$
\begin{equation}
\begin{aligned} \, &\sup_{1\leqslant r<R<+\infty} \ell^{\operatorname{rh}}_{\nu-\mu}(r,R) \stackrel{(2.3)}{\leqslant} \sup_{1\leqslant r<R<+\infty} \bigl(\ell_{\nu}(r,R)-\ell^{\operatorname{rh}}_{\mu}(r,R)\bigr) \nonumber \\ &\qquad\leqslant \sup_{1\leqslant r<R<+\infty} \bigl(\bigl(\ell_{\nu}(r,R)- \ell_{\mu}(r,R)\bigr) + |\ell_{\mu}(r,R)-\ell^{\operatorname{rh}}_{\mu}(r,R)|\bigr) \stackrel{(5.6)}{<}+\infty, \end{aligned}
\end{equation}
\tag{5.11}
$$
where (5.10) was used at the end. Similarly, from (5.6) and (5.10) we have
$$
\begin{equation}
\sup_{1\leqslant r<R<+\infty} \ell^{\operatorname{lh}}_{\nu-\mu}(r,R)<+\infty.
\end{equation}
\tag{5.12}
$$
An appeal to (5.7) gives $\pm b\notin \operatorname{supp} (\nu+\mu)$, and so by Remark 8 it is possible to be confined with balayage of genus $1$ when implementing steps [b2] and [b4]
Let us proceed with steps [b1], [b2], where in step [b2] we apply Proposition 5 to the distributions of masses $\nu_{\vec{-b}}\lfloor_{\mathbb{C}_{\mathrm{rh}}}$ and $\mu_{\vec{-b}}\lfloor_{\mathbb{C}_{\mathrm{rh}}}$ in place of $\nu$ and $\mu$, respectively, for which by (5.7) their supports lie in $\{z\in \mathbb{C}\mid \operatorname{Re} z > a|z|\}$ and (4.11) holds, and condition (4.11) is secured by (5.11). So, at step [b2] we get a distribution of masses $\alpha^{\operatorname{rh}}$ of finite upper density with support on $\mathbb{R}^+\setminus [0,r_0)$ for $r_0>0$, a distribution of masses $\beta^{\operatorname{rh}}$ with support on $i\mathbb{R}$, and a number $c^{\operatorname{rh}}\in \mathbb{R}^+$ such that
$$
\begin{equation}
\bigl(\nu_{\vec{-b}}\lfloor_{\mathbb{C}_{\mathrm{rh}}}+ \alpha^{\operatorname{rh}} -\mu_{\vec{-b}} \lfloor_{\mathbb{C}_{\mathrm{rh}}}\bigr)^{\operatorname{bal}^1_{\mathrm{rh}}} +\beta^{\operatorname{rh}} \stackrel{(4.12)}{=} c^{\operatorname{rh}}\mathfrak m_1\lfloor_{i\mathbb{R}},
\end{equation}
\tag{5.13}
$$
$$
\begin{equation}
\nu_{\vec{-b}}\lfloor_{\mathbb{C}_{\mathrm{rh}}}+\alpha^{\operatorname{rh}} + \beta^{\operatorname{rh}}-\mu_{\vec{-b}}\lfloor_{\mathbb{C}_{\mathrm{rh}}} \quad \text{satisfies the Lindel}\unicode{x00F6}\text{f condition}.
\end{equation}
\tag{5.14}
$$
After this, we use [b3], [b4], where, at step [b4], we again apply Proposition 5 in the $i\mathbb{R}$-reflection symmetric form to the distributions of masses $\nu_{\vec{b}}\lfloor_{\mathbb{C}_{\mathrm{lh}}}$ and $\mu_{\vec{b}}\lfloor_{\mathbb{C}_{\mathrm{lh}}}$ in place of $\nu$ and $\mu$. That their supports lie in the angle $\{z\in \mathbb{C}\mid \operatorname{Re} z <- a|z|\}$ and (4.11) holds with $\ell^{\operatorname{lh}}$ instead of $\ell^{\operatorname{rh}}$ follows from (5.7) and (5.12). This gives us a distribution of masses $\alpha^{\operatorname{lh}}$ of finite upper density with $\operatorname{supp} \alpha^{\operatorname{lh}} \subset -\mathbb{R}^+\setminus [0,r_0)$ for some $r_0>0$, a distribution of masses $\beta^{\operatorname{lh}}$ with $\operatorname{supp} \beta^{\operatorname{lh}}\subset i\mathbb{R}$, and a number $c^{\operatorname{lh}}\in \mathbb{R}^+$ such that
$$
\begin{equation}
\bigl(\nu_{\vec{b}}\lfloor_{\mathbb{C}_{\mathrm{lh}}}+ \alpha^{\operatorname{lh}} -\mu_{\vec{b}} \lfloor_{\mathbb{C}_{\mathrm{lh}}}\bigr)^{\operatorname{bal}^1_{\mathrm{lh}}} +\beta^{\operatorname{lh}} \stackrel{(4.12)}{=} c^{\operatorname{lh}}\mathfrak m_1\lfloor_{i\mathbb{R}},
\end{equation}
\tag{5.15}
$$
$$
\begin{equation}
\nu_{\vec{b}}\lfloor_{\mathbb{C}_{\mathrm{lh}}}+\alpha^{\operatorname{lh}} +\beta^{\operatorname{lh}} -\mu_{\vec{b}}\lfloor_{\mathbb{C}_{\mathrm{lh}}} \quad \text{satisfies the Lindel}\unicode{x00F6}\text{f condition}.
\end{equation}
\tag{5.16}
$$
If $c^{\operatorname{rh}}\geqslant c^{\operatorname{lh}}$, then we can add the distribution of masses $(c^{\operatorname{rh}}- c^{\operatorname{lh}})\mathfrak m_1\lfloor_{i\mathbb{R}}$ of finite upper density satisfying the Lindelöf condition to the right-hand side (5.15) and to $\beta^{\operatorname{lh}}$, keeping the same notation $\beta^{\operatorname{lh}}$ for the sums $\beta^{\operatorname{lh}}+(c^{\operatorname{rh}}- c^{\operatorname{lh}})\mathfrak m_1\lfloor_{i\mathbb{R}}$. It is clear that (5.16) is preserved in this way, and the right-hand sides of (5.13) and (5.15) will contain $c^{\operatorname{lh}}=c^{\operatorname{rh}}$. We proceed similarly for $c^{\operatorname{rh}}< c^{\operatorname{lh}}$ in relation to (5.13), (5.14), which gives us $c^{\operatorname{rh}}=c^{\operatorname{lh}}$. Hence we can always choose
At step [b5], we get the required distribution of masses $\alpha:=\alpha^{\operatorname{rh}}_{\vec{b}}+ \alpha^{\operatorname{lh}}_{\vec{-b}}$ of finite upper density with support on $\mathbb{R}\setminus [-b,b]$ and distributions of masses $\beta_+:=\beta_{\vec{b}}^{\operatorname{rh}}$ and $\beta_-:=\beta_{\vec{-b}}^{\operatorname{lh}}$ of finite upper density with supports, respectively, on the lines $b+i\mathbb{R}$ and $-b+i\mathbb{R}$. In this case, the balayages at steps [b2] and [b4] give the balayage
$$
\begin{equation}
(\nu+\alpha -\mu)^{\operatorname{Bal}^1_{b}}+\beta_++\beta_-= (\nu+\alpha+\beta_++\beta_- -\mu)^{\operatorname{Bal}^1_b}
\end{equation}
\tag{5.18}
$$
of finite order from (5.3) with support on the pair of lines $\pm b+i\mathbb{R}$, where equality (5.18) follows from the equality $(\beta_++\beta_-)^{\operatorname{Bal}^1_{b}}=\beta_++\beta_-$ for distributions with support on $\overline{\operatorname{str}}_b$. Now (5.8) is secured by (5.18) and (5.13), (5.15), and (5.17). From (5.14) and (5.16) it follows that the distribution of charges $\nu+\alpha+\beta_++\beta_--\mu$ satisfies the Lindelöf condition. By adding this distribution of charges to the distribution of masses $\mu$ satisfying the Lindelöf condition we get the distribution of masses $\nu+\alpha+\beta_++\beta_-$, which also obeys the Lindelöf condition. This proves Proposition 7. 5.2. Shifts and balayage of a $\delta$-subharmonic functionLet $u$ be a function on $\mathbb{C}$ and $w\in \mathbb{C}$. Similarly to the $w$-shift (5.1) of distribution of charges, the $w$-shift $u_{\vec{w}}$ of a function is defined by
$$
\begin{equation}
u_{\vec{w}}\colon z\underset{z\in \mathbb{C}}{\longmapsto} u(z-w).
\end{equation}
\tag{5.19}
$$
The $w$-shift (5.19) of a $\delta$-subharmonic function $\mathcal U\not\equiv \pm \infty$ on $\mathbb{C}$ results in the $w$-shift
$$
\begin{equation}
\frac{1}{2\pi}\Delta (\mathcal U_{\vec{w}})\stackrel{(5.1)}{=} \biggl(\frac{1}{2\pi}\Delta \mathcal U\biggr)_{\vec{w}}
\end{equation}
\tag{5.20}
$$
of the Riesz distribution of charges of this function. For a $\delta$-subharmonic function $\mathcal U\not\equiv \pm\infty$ and $b\in \mathbb{R}^+$, by a $\delta$-subharmonic balayage on the strip $\overline{\operatorname{str}}_b$ of the function S$\mathcal U$ we will mean any $\delta$-subharmonic function $\mathcal U^{\operatorname{Bal}_b}$ equal to $\mathcal U$ on the vertical strip $\overline{\operatorname{str}}_b$ of width $2b$ from (1.5) outside some polar set and, at the same time, harmonic on $\mathbb{C}\setminus \overline{\operatorname{str}}_b$. Proposition 8. Let $b\in \mathbb{R}^+$ and let a $\delta$-subharmonic function $\mathcal U\not\equiv \pm \infty$ with Riesz distribution of charges (3.40) of finite upper density be representable as the difference of subharmonic functions of order $\leqslant 1$. Then there exists a $\delta$-subharmonic balayage $\mathcal U^{\operatorname{Bal}_b}$ on $\overline{\operatorname{str}}_b$ of the function $\mathcal U$ with Riesz distribution of charges Proof. The construction of the function $\mathcal U^{\operatorname{Bal}_b}$ is in five successive steps [B1]–[B5], each successive step is applied to the $\delta$-subharmonic function obtained at the previous step.
[B1] The $(-b)$-shift $\mathcal U_{\vec{-b}}$ of the function $\mathcal U$ with the equality $\frac{1}{2\pi}\Delta (\mathcal U_{\vec{-b}}) \stackrel{(5.20)}{=}(\varDelta_\mathcal U)_{\vec{-b}}$ and with obvious preservation, for $\mathcal U_{\vec{-b}}$, of the representation in the form of the difference of subharmonic functions of order $\leqslant 1$. [B2] The $\delta$-subharmonic balayage $\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}}$ of the function $\mathcal U_{\vec{-b}}$ from $\mathbb{C}_{\mathrm{rh}}$ onto $\mathbb{C}_{\overline{\mathrm{lh}}}$ via Proposition 3 with due account of equality (3.41) in the form $\frac{1}{2\pi}\Delta \bigl(\mathcal{U}^{\operatorname{Bal}_{\mathrm{rh}}}_{\vec{-b}}\bigr) \stackrel{(3.41)}{=} (\varDelta_\mathcal U)_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}}$ and with representation of this $\delta$-subharmonic balayage as the difference of subharmonic functions of order $\leqslant 1$. [B3] The $2b$-shift $\bigl(\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}} \bigr)_{\vec{2b}}$ of $\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}}$, $\frac{1}{2\pi}\Delta \bigl(\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}} \bigr)_{\vec{2b}} \stackrel{(5.20)}{=} \bigl((\varDelta_\mathcal U)_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}} \bigr)_{\vec{2b}}$, with obvious preservation for $(\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}})_{\vec{2b}}$ of the representation as the difference of subharmonic functions of order $\leqslant 1$. [B4] An application of the reflection symmetry with respect to $i\mathbb{R}$ and a balayage from $\mathbb{C}_{\mathrm{rh}}$ onto $\mathbb{C}_{\overline{\mathrm{lh}}}$ with reverse reflection symmetry allows us to determine a $\delta$-subharmonic balayage from $\mathbb{C}_{\mathrm{lh}}$ onto $\mathbb{C}_{\overline{\mathrm{rh}}}$ with natural upper indexation of the form $^{\operatorname{Bal}_{\mathrm{lh}}}$; an application of this balayage via Proposition 3 to the function $\bigl(\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}} \bigr)_{\vec{2b}}$ with due account of (3.41) gives
$$
\begin{equation*}
\frac{1}{2\pi}\Delta \bigl(\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}} \bigr)_{\vec{2b}}^{\operatorname{Bal}_{\mathrm{lh}}} \stackrel{(3.41)}{=}\bigl( (\varDelta_\mathcal U)_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}} \bigr)_{\vec{2b}}^{\operatorname{bal}^{01}_{\mathrm{lh}}};
\end{equation*}
\notag
$$
we also have a representation of $\bigl(\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}} \bigr)_{\vec{2b}}^{\operatorname{Bal}_{\mathrm{lh}}}$ as the difference of subharmonic functions of order $\leqslant 1$.
[B5] The $(-b)$-shift $\bigl(\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}})_{\vec{2b}}^{\operatorname{Bal}_{\mathrm{lh}}}\bigr)_{\vec{-b}}$ of the function $\bigl(\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}} \bigr)_{\vec{2b}}^{\operatorname{Bal}_{\mathrm{lh}}}$ with
$$
\begin{equation}
\frac{1}{2\pi}\Delta \Bigl(\bigl(\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}} \bigr)_{\vec{2b}}^{\operatorname{Bal}_{\mathrm{lh}}}\Bigr)_{\vec{-b}} \stackrel{(5.20)}{=}\biggl(\Bigl( (\varDelta_\mathcal U)_{\vec{-b}}^{\operatorname{bal}^{01}_{\mathrm{rh}}} \Bigr)_{\vec{2b}}^{\operatorname{bal}^{01}_{\mathrm{lh}}}\biggr)_{\vec{-b}} \stackrel{(5.3)}{=} \varDelta_\mathcal{U}^{\operatorname{Bal}^{01}_b},
\end{equation}
\tag{5.22}
$$
where, by construction, $\bigl((\mathcal U_{\vec{-b}}^{\operatorname{Bal}_{\mathrm{rh}}} )_{\vec{2b}}^{\operatorname{Bal}_{\mathrm{lh}}}\bigr)_{\vec{-b}} = \mathcal{U}^{\operatorname{Bal}_b}$ is some $\delta$-subharmonic balayage of the $\delta$-subharmonic function $\mathcal U$ onto $\overline{\operatorname{str}}_b$, which can be represented as the difference of subharmonic functions of order $\leqslant 1$, (5.22) is precisely equality (5.21). This proves Proposition 8. Remark 9. If, under the conditions of Proposition 8, the function $\mathcal U$ is harmonic in two open semicircles $b+r_0\mathbb{D}\cap \mathbb{C}_{\mathrm{rh}}$ and $-b+r_0\mathbb{D}\cap \mathbb{C}_{\mathrm{lh}}$ for some $r_0>0$, then by the variant (5.5) of Remark 8 the right-hand side in (5.21) can be replaced by the balayage $\varDelta_\mathcal U^{\operatorname{Bal}^{1}_b}$ of genus $1$ onto $\overline{\operatorname{str}}_{b}$, that is, § 6. Balayage of a subharmonic function of finite type onto the union of a vertical strip and the real axis Proposition 9. Let $M\not\equiv -\infty$ be a subharmonic function of finite type with Riesz mass distribution $\varDelta_M$. Then, for any $s\in \mathbb{R}^+$, there exists a subharmonic function $M_{\mathbb{R}}$ of finite type with Riesz distribution of masses $\varDelta_{M_{\mathbb{R}}}=\frac{1}{2\pi}\Delta M_{\mathbb{R}}$ concentrated on $\mathbb{R}\setminus [-s,s]$ and such that The key role in the proof will be played by the following fairly general lemma. Lemma 2 (see [48], the main theorem, [21], Proposition 2.1, and [9], Theorem 8). Let $S$ be a closed system of rays on $\mathbb{C}$ with one common vertex and let $p\in \mathbb{R}^+$. Suppose that the opening of any open angle complementary to $S$ (that is, the connected component in $\mathbb{C}\setminus S$) is $<\pi /p$. Then, for every subharmonic function $u\not\equiv -\infty$ of finite type for order $p$, there exists a subharmonic function $u^{\operatorname{bal}}\geqslant u$ on $\mathbb{C}$ of finite type of order $p$ which is equal to $u$ on each ray of $S$ and is harmonic on each angle complementary to $S$. The function $u^{\operatorname{bal}}$ is the balayage of the function $u$ from the open set $\mathbb{C}\setminus S$ onto the system of rays $S$. In this case, the system of rays $S$ is also considered as a closed subset of $\mathbb{C}$ of all points lying on rays from $S$. Let us first consider an infinite closed system of rays $S_s^-$ consisting of the union of the ray $[s,+\infty)\subset \mathbb{R}$ with all rays from the closed half-plane $\{z\in \mathbb{C}\mid \operatorname{Re} z\leqslant s\}$ with common vertex at $s\in \mathbb{R}$ and with two additional open right angles $\pi/2$. Applying Lemma 2 with $p:=1$ to the function $M$ in place of $u$ and to the system of rays $S_s^-$, we obtain a subharmonic function $M^{\operatorname{bal}}\geqslant M$ of finite type on $\mathbb{C}$ which is harmonic on $\mathbb{C}\setminus S_s^-$ and is equal to $M$ on $S_s^-$. Now let $S_{-s}^+$ be an infinite closed system of rays consisting of the union of the ray $(-\infty, -s]\subset \mathbb{R}$ with all rays from the closed half-plane $\{z\in \mathbb{C}\mid \operatorname{Re} z\leqslant -s\}$ with common vertex at $-s\in \mathbb{R}$ and also with two additional open right angles. Applying Lemma 2 with $p:=1$ to the function $M^{\operatorname{bal}}$ in place of $u$ and to the system of rays $S_{-s}^+$, we obtain a subharmonic function $(M^{\operatorname{bal}})^{\operatorname{bal}}\geqslant M^{\operatorname{bal}} \geqslant M$ of finite type on $\mathbb{C}$, which coincides with the function $M$ on the intersection $S_s^-\cap S_{-s}^+=\mathbb{R}\cup \overline{\operatorname{str}}_s$ and is harmonic outside this intersection. By the first part of the Weierstrass–Hadamard–Lindelöf–Brelot theorem, the Riesz distribution of masses $\frac{1}{2\pi}\Delta (M^{\operatorname{bal}})^{\operatorname{bal}}$ of finite upper density satisfying the Lindelöf condition and concentrated on $\mathbb{R}\cup \overline{\operatorname{str}}_s$ can be divided into the sum of the restriction $\mu_{s}:=\frac{1}{2\pi}\Delta (M^{\operatorname{bal}})^{\operatorname{bal}} \lfloor_{\overline{\operatorname{str}}_s}$ to the closed strip $\overline{\operatorname{str}}_s$ and the remaining part of $\mu_{\mathbb{R}}$ concentrated on $\mathbb{R}\setminus [-s,s]$. Each of these two distributions of masses $\mu_{s}$ and $\mu_{\mathbb{R}}$ is, obviously, of finite upper density, and their sum $\mu_s+\mu_{\mathbb{R}}= \frac{1}{2\pi}\Delta (M^{\operatorname{bal}})^{\operatorname{bal}}$ satisfies the Lindelöf condition. Hence, since $\mu_{\mathbb{R}}$ is concented on $\mathbb{R}$, the distribution of masses $\mu_s$ satisfies the Lindelöf $i\mathbb{R}$ condition
$$
\begin{equation}
\sup_{r\geqslant 1}\biggl|\int_{1< |z|\leqslant r} \operatorname{Im} \frac{1}{z}\,d \mu_{s}(z)\biggr|\stackrel{(3.20)}{<}+\infty.
\end{equation}
\tag{6.5}
$$
The distribution of masses $\mu_{s}$ also satisfies the Lindelöf $\mathbb{R}$-condition (3.18), since
$$
\begin{equation}
\begin{aligned} \, &\sup_{r>1}\biggl|\int_{1< |z|\leqslant r} \operatorname{Re} \frac{1}{z}\,d \mu_{s}(z)\biggr|\leqslant \int_{|z|> 1} \biggl|\operatorname{Re} \frac{1}{ z}\biggr|\,d \mu_{s}(z) \nonumber \\ &\qquad\stackrel{(1.13)}{\leqslant} |s|\int_1^{+\infty} \frac{d \mu^{\mathrm{rad}}_{s}(t)}{t^2} \leqslant 2|s|\int_1^{+\infty} \frac{\mu^{\mathrm{rad}}_s(t)}{t^3}\,d t<+\infty \end{aligned}
\end{equation}
\tag{6.6}
$$
for the distribution of masses $\mu_{s}$ of finite upper density. Therefore, the distribution of masses $\mu_{s}$ satisfies the Lindelöf condition (3.21), as does the distribution of masses $\mu_\mathbb{R}$, which is the difference of two distributions of masses satisfying the Lindelöf condition (3.21). By the second part of the Weierstrass–Hadamard–Lindelöf–Brelot theorem, there exists a subharmonic function $M_{\mathbb{R}}$ of finite type with Riesz distribution of masses $\frac{1}{2\pi}\Delta M_{\mathbb{R}}=\mu_{\mathbb{R}}$. The subharmonic function $M_{s}:=(M^{\operatorname{bal}})^{\operatorname{bal}} -M_{\mathbb{R}}$ with Riesz distribution of masses $\frac{1}{2\pi}\Delta M_{s}=\mu_{s}$ is also of finite type. By the above construction, all the requirements of Proposition 9 to the functions $M_\mathbb{R}$ and $M_{s}$, including (6.2)–(6.4), are satisfied, except (6.1). Lemma 3 (see [28], Proposition 4.1, formula (4.19)). For each subharmonic function $u\not\equiv -\infty$ of finite type with Riesz distribution of masses $\varDelta_u$, if Applying inequality (6.8) in this lemma to the $(-s)$-shift $M_{\vec{-s}}$ of the function $M$ by $-s$, as defined in (5.19), we get
$$
\begin{equation}
\sup_{1\leqslant r<R<+\infty} \bigl|\ell^{\operatorname{rh}}_{\varDelta_{M_{\vec{-s}}}}(r,R)- J_{i\mathbb{R}}(r,R;M_{\vec{-s}})\bigr|<+\infty,
\end{equation}
\tag{6.11}
$$
where $\varDelta_{M_{\vec{-s}}}$ is the $(-s)$-shift of the mass distribution from (5.1). By applying $(M_{\mathbb{R}}+M_{s})_{\vec{-s}}$ to this shift, we obtain
$$
\begin{equation}
\sup_{1\leqslant r<R<+\infty} \bigl|J_{i\mathbb{R}}\bigl(r,R;(M_{\mathbb{R}}+ M_{s})_{\vec{-s}}\bigr)-\ell^{\operatorname{rh}}_{\varDelta_{(M_{\mathbb{R}}+ M_{s})_{\vec{-s}}}}(r,R) \bigr|<+\infty.
\end{equation}
\tag{6.12}
$$
By (6.3), the functions $(M_{\mathbb{R}}+M_{s})_{\vec{-s}}$ and $M_{\vec{-s}}$ coincide on $i\mathbb{R}$. Adding (6.11), (6.12), and estimating the sums of the suprema of the absolute values, we obtain
$$
\begin{equation}
\sup_{1\leqslant r<R<+\infty} \bigl|\ell^{\operatorname{rh}}_{\varDelta_{M_{\vec{-s}}}}(r,R)- \ell^{\operatorname{rh}}_{\varDelta_{(M_{\mathbb{R}}+M_{s})_{\vec{-s}}}} (r,R) \bigr|<+\infty.
\end{equation}
\tag{6.13}
$$
But the Riesz distribution of masses $\varDelta_{(M_s)_{\vec{-s}}}$ of the shift $(M_s)_{\vec{-s}}$ of the function $M_s$ is concentrated on $ \mathbb{C}_{\overline{\mathrm{lh}}}$. So, by the definition of the right logarithmic measure,
$$
\begin{equation*}
\ell^{\operatorname{rh}}_{\varDelta_{(M_{\mathbb{R}}+M_s)_{\vec{-s}}}} \stackrel {(2.1)}{=} \ell^{\operatorname{rh}}_{\varDelta_{(M_{\mathbb{R}})_{\vec{-s}}}}.
\end{equation*}
\notag
$$
Hence by (6.13) we have
$$
\begin{equation}
\sup_{1\leqslant r<R<+\infty} \bigl|\ell^{\operatorname{rh}}_{\varDelta_{M_{\vec{-s}}}}(r,R)- \ell^{\operatorname{rh}}_{\varDelta_{(M_{\mathbb{R}})_{\vec{-s}}}} (r,R) \bigr|<+\infty.
\end{equation}
\tag{6.14}
$$
Since the upper mass distribution $\varDelta_M$ has finite density, we have by Proposition 6
$$
\begin{equation*}
\begin{gathered} \, \sup_{1\leqslant r<R<+\infty}\bigl|\ell^{\operatorname{rh}}_{\varDelta_M}(r,R)- \ell^{\operatorname{rh}}_{\varDelta_{M_{\vec{-s}}}}(r,R)\bigr|<+\infty, \\ \sup_{1\leqslant r<R<+\infty} \bigl|\ell^{\operatorname{rh}}_{\varDelta_{(M_{\mathbb{R}})_{\vec{-s}}}}(r,R) -\ell^{\operatorname{rh}}_{\varDelta_{M_{\mathbb{R}}}} (r,R)\bigr|<+\infty. \end{gathered}
\end{equation*}
\notag
$$
Adding these relations with (6.14) and estimating from below the sums of the suprema of the absolute values by the supremum of the absolute values of the sum, we arrive at (6.1).
§ 7. Proof of the implication II $\Rightarrow$ I of the main theoremProposition 10. Let $\mu$ and $\nu$ be mass distributions of finite upper density. Then the following two assertions are equivalent. 1. There is an unbounded sequence $(r_n)_{n\in \mathbb{N}}$ in $\mathbb{R}^+\setminus 0$ increasing no faster than a geometric progression in the sense that and such that
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_{\nu}(r_n,r_N) -\ell_{\mu}(r_n,r_N)\bigr) <+\infty.
\end{equation}
\tag{7.2}
$$
2. For any $r_0\in \mathbb{R}^+\setminus 0$,
$$
\begin{equation}
\sup_{r_0\leqslant r<R<+\infty} \bigl(\ell_{\nu}(r,R)- \ell_{\mu}(r,R)\bigr)<+\infty.
\end{equation}
\tag{7.3}
$$
In addition, if $\mu$ is the Riesz distribution mass of a subharmonic function $M\not\equiv -\infty$ of finite type, then the previous assertions 1, 2 are equivalent to each of the following two assertions. 3. There a sequence $(r_n)_{n\in \mathbb{N}}$ as in assertion 1 with property (7.1), for which, in notation (6.7),
$$
\begin{equation}
\limsup_{r_N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_{\nu}(r_n,r_N)- J_{i\mathbb{R}}(r_n,r_N;M)\bigr)<+\infty.
\end{equation}
\tag{7.4}
$$
4. Relation (2.41) is satisfied. Proof. The equivalences 1 $\Leftrightarrow$ 3 and 1 $\Leftrightarrow$ 4 follow from (6.10) in Lemma 3. The implication 2 $\Rightarrow$ 1 is obvious. If assertion 1 holds, then by (7.1) there exists a number $A>1$ such that $r_{n+1}\leqslant Ar_n$ for all $n\in \mathbb{N}$, where we can consider an arbitrary $r_0\in (0,r_1]$, and also by (7.2) there exist a number $B>0$ and a number $N_0\in \mathbb{N}$ such that for every $N> N_0$ and any $n<N$. For all $0\leqslant n<N\leqslant N_0$,
$$
\begin{equation*}
\ell_{\nu}(r_n,r_N)\leqslant \int_{r_0}^{r_{N_0}}\Bigl|\operatorname{Re} \frac{1}{z}\Bigr|\,d \nu(z)\leqslant B_0,
\end{equation*}
\notag
$$
where $B_0$ is independent of $n<N$. Thus, for a sufficiently large $B>0$, inequalities (7.5) are satisfied for all integer $N>n\geqslant 0$. For $r_0\leqslant r<R$, let $n\in \mathbb{N}$ and $N\geqslant n$ be such that $r\in (r_{n},r_{n+1}]$ and $R\in (r_N,r_{N+1}]$. Hence
$$
\begin{equation*}
\begin{aligned} \, \ell_{\nu}(r,R) &\leqslant \ell_{\nu}(r_n,r_{n+1})+\ell_{\nu}(r_{n+1},r_N)+ \ell_{\nu}(r_N, r_{N+1}) \\ &\!\!\!\stackrel{(7.5)}{\leqslant} \ell_{\nu}(r_n,Ar_n) + \ell_{\mu}(r_n,r_N)+B+\ell_{\nu}(r_N,Ar_N) \\ &\leqslant \frac{\nu^{\mathrm{rad}}(Ar_n)}{r_n}+ \ell_{\mu}(r,R)+B+ \frac{\nu^{\mathrm{rad}}(Ar_N)}{r_N}. \end{aligned}
\end{equation*}
\notag
$$
Now (7.3) follows since the distribution of masses $\nu$ has finite upper density. This proves Proposition 10. Proof of the implication II $\Rightarrow$ I. From (2.15) of assertion II in notation (6.7) we have, for some $C\in \mathbb{R}$,
$$
\begin{equation}
\ell_{\nu}(2^n,2^{n+1})\leqslant J_{i\mathbb{R}}(2^n, 2^{n+1};M)+C\leqslant \operatorname{type}[M]+1+C
\end{equation}
\tag{7.6}
$$
for all sufficiently large $n\in \mathbb{N}$. Next, from the limit relation, under conditions (2.13) of the main theorem (or condition [$ \boldsymbol\nu $] with (2.23), which follows from this result by Remark 1, we have, as in (5.9), the lower estimates
$$
\begin{equation*}
\ell_{\nu}(2^n,2^{n+1})\geqslant \ell_\nu^{\operatorname{rh}}(2^n, 2^{n+1}) \stackrel{(5.9)}{\geqslant} a2^{-n-1}\bigl(\nu \lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}^{\mathrm{rad}}(2^{n+1}) -\nu\lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}^{\mathrm{rad}} (2^{n})\bigr),
\end{equation*}
\notag
$$
whence by (7.6) we have
$$
\begin{equation*}
\nu\lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}^{\mathrm{rad}}(2^{n+1}) -\nu\lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}^{\mathrm{rad}}(2^{n}) \leqslant (\operatorname{type}[M]+1+C)\frac{1}{a} 2^{n+1}
\end{equation*}
\notag
$$
for sufficiently large $n\in \mathbb{N}$. This means that $\nu\lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}$ is the distribution of masses with finite upper density. A similar analysis produces the same result for the restriction $\nu\lfloor_{\mathbb{C}_{\mathrm{lh}}}$. Hence, from (2.15) and the limit relation in (2.13) it follows that the distribution of masses $\nu$ has finite upper density.
In view of the equivalence 3 $\Leftrightarrow$ 2 of Proposition 10, for the binary sequence $r_n:=2^n$ in (7.1), relation (2.15) of assertion II of the main theorem implies that
$$
\begin{equation}
\sup_{1\leqslant r<R<+\infty} \bigl(\ell_{\nu}(r,R)- \ell_{\varDelta_M}(r,R)\bigr)<+\infty
\end{equation}
\tag{7.7}
$$
with Riesz distribution of masses $\varDelta_M\stackrel{(1.12)}{=}\frac{1}{2\pi}\Delta M$ of the subharminc function $M$. Let us apply Proposition 9 to the balayage of the function $M$ onto the union of $\mathbb{R}$ with the closed vertical strip $\overline{\operatorname{str}}_{s}$ of width $2s$, where $s$ is defined by the equality in (2.13) in the main theorem. Under the notation of Proposition 9, using (7.7) and (6.1), we obtain
$$
\begin{equation*}
\begin{aligned} \, &\sup_{1\leqslant r<R<+\infty} \bigl(\ell_{\nu}(r,R) -\ell_{\varDelta_{M_{\mathbb{R}}}}(r,R)\bigr) \leqslant \sup_{1\leqslant r<R<+\infty} \bigl(\ell_{\nu}(r,R)-\ell_{\varDelta_M}(r,R)\bigr) \\ &\qquad+\sup_{1\leqslant r<R<+\infty}|\ell_{\varDelta_M}(r,R) -\ell_{\varDelta_{M_{\mathbb{R}}}}(r,R)| \stackrel{(7.7),(6.1)}{<} +\infty. \end{aligned}
\end{equation*}
\notag
$$
The left- and right-hand parts of these inequalities can be written as
$$
\begin{equation}
\sup_{1\leqslant r<R<+\infty} \bigl(\ell_{\nu}(r,R)-\ell_{\mu}(r,R)\bigr)<+\infty,
\end{equation}
\tag{7.8}
$$
where for the Riesz distribution of masses of the subharmonic function $M_{\mathbb{R}}$ we use the notation
$$
\begin{equation}
\mu:=\varDelta_{M_{\mathbb{R}}}, \qquad \operatorname{supp} \mu=\operatorname{supp} \varDelta_{M_{\mathbb{R}}} \subset \mathbb{R}\setminus (-s,s).
\end{equation}
\tag{7.9}
$$
Consider an arbitrary fixed number By (7.9), in addition to (7.8), which coincides with condition (5.6) of Proposition 7, we also have condition (5.7) of Proposition 7 with $b\in (0,s)$, which follows from (7.9), Hence Proposition 7 is verified. The notation of the entities from this proposition will be used below.By the Weierstrass–Hadamard–Lindelöf–Brelot theorem, for the distribution of masses $\nu+\alpha+\beta_++\beta_-$ of finite upper density satisfying the Lindelöf condition, there exists a subharmonic function $u$ of finite type with Riesz distribution of masses
$$
\begin{equation}
\varDelta_u=\nu+\alpha+\beta_++\beta_-\geqslant \nu.
\end{equation}
\tag{7.11}
$$
From (7.9), the construction of $\alpha$ and $\beta_{\pm}$ in Proposition 7, and conditions (2.13) on $\nu$ (in the form of condition [$ \boldsymbol\nu $] with (2.23)), using Remark 1 we have
$$
\begin{equation}
\operatorname{supp} (\nu+\alpha-\mu)\subset \mathbb{C}\setminus (\overline {\mathrm X}_a\cup \overline{\operatorname{str}}_{b}), \qquad \operatorname{supp} \beta_{\pm}\subset \pm b+i\mathbb{R}.
\end{equation}
\tag{7.12}
$$
Consider the $\delta$-subharmonic function
$$
\begin{equation}
\mathcal{U}:=u-M_{\mathbb{R}}, \qquad \varDelta_\mathcal{U}:= \frac{1}{2\pi}\Delta \mathcal{U}=\nu+\alpha+\beta_++\beta_- -\mu.
\end{equation}
\tag{7.13}
$$
This function is the difference of subharmonic functions of finite type. By Proposition 8, there exists a $\delta$-subharmonic balayage $\mathcal U^{\operatorname{Bal}_b}$ on $\overline{\operatorname{str}}_b$ of the function $\mathcal U$ with Riesz distribution of charges (5.21). The function $\mathcal U^{\operatorname{Bal}_b}$ is represented as the difference (3.42) of subharmonic functions of order $\leqslant 1$ outside some polar set. By (7.12) and (7.13), the function $\mathcal U$ is harmonic in two open semidiscs $b+r_0\mathbb{D}\cap \mathbb{C}_{\mathrm{rh}}$ and $-b+r_0\mathbb{D}\cap \mathbb{C}_{\mathrm{lh}}$ for $r_0:=s-b>0$. Therefore, by Remark 9, the right-hand side in (5.21) can be replaced, as in (5.23), by the balayage $\varDelta_\mathcal U^{\operatorname{Bal}^{1}_b}$ of genus $1$ onto $\overline{\operatorname{str}}_{b}$, which by construction has the form
$$
\begin{equation*}
\begin{aligned} \, \varDelta_\mathcal{U}^{\operatorname{Bal}^{1}_b} &=(\nu+\alpha+\beta_++ \beta_--\mu)^{\operatorname{Bal}^{1}_b} \\ &\!\!\!\stackrel{(5.8)}{=}(\nu+\alpha -\mu)^{\operatorname{Bal}^1_{b}} +\beta_++\beta_- \stackrel{(5.8)}{=}c\mathfrak m_1 \lfloor_{b+i\mathbb{R}} +c\mathfrak m_1\lfloor_{-b+i\mathbb{R}}. \end{aligned}
\end{equation*}
\notag
$$
The right-hand side here is explicitly written as the sum of a pair of linear Lebesgue measures multiplied by $c\in \mathbb{R}^+$ with support on $\pm b+i\mathbb{R}$. Obviously, this sum satisfies the Lindelöf condition. Hence, by the second part of the Weierstrass–Hadamard–Lindelöf–Brelot theorem, the function $\mathcal U^{\operatorname{Bal}_b}$ is the sum of the subharmonic function
$$
\begin{equation}
2\pi c (\operatorname{Re} z-b)^++2\pi c (\operatorname{Re} z+b)^- \underset{z\in \mathbb{C}}{=} \begin{cases} 2\pi c (\operatorname{Re} z-b)^+ &\text{for }\operatorname{Re} z>b, \\ 0 &\text{for }|{\operatorname{Re} z}|\leqslant b, \\ 2\pi c (\operatorname{Re} z+b)^- &\text{for }\operatorname{Re} z<-b \end{cases}
\end{equation}
\tag{7.14}
$$
with some harmonic polynomial $h$ of degree $\deg h\leqslant 1$. From these constructions, by the definition of the $\delta$-subharmonic balayage onto the strip $\overline{\operatorname{str}}_b$, we have
$$
\begin{equation*}
\mathcal U^{\operatorname{Bal}_b}(z)\stackrel{(7.13)}{\equiv} u(z)- M_{\mathbb{R}}(z) \quad\text{for all }z\in \overline{\operatorname{str}}_b.
\end{equation*}
\notag
$$
Hence, since $\mathcal U^{\operatorname{Bal}_b}$ is the sum of function (7.14) and $h$, we obtain
$$
\begin{equation}
(u-M_{\mathbb{R}}-h)(z)= \mathcal U^{\operatorname{Bal}_b}(z)-h(z) \stackrel{(7.14)}{\equiv}0 \quad\text{for all } z\in \overline{\operatorname{str}}_b,
\end{equation}
\tag{7.15}
$$
but outside some polar set. Thus, we have constructed a subharmonic function $u-h$ of finite type such that
$$
\begin{equation}
(u-h)(z)\stackrel{(7.15)}{\equiv} M_{\mathbb{R}}(z) \quad\text{for all }z\in \overline{\operatorname{str}}_b,
\end{equation}
\tag{7.16}
$$
with Riesz distribution of masses
$$
\begin{equation}
\varDelta_{u-h}=\frac{1}{2\pi}(\Delta u-\Delta h)= \frac{1}{2\pi}\, \Delta u \stackrel{(7.11)}{=}\nu+\alpha+\beta_++\beta_-\geqslant \nu.
\end{equation}
\tag{7.17}
$$
Consider the subharmonic function $U:=(u-h)+M_s$, where $M_s$ is the subharmonic function of finite type defined in Proposition 9. This function is also of finite type with Riesz distribution of masses $\varDelta_U\geqslant \varDelta_{u-h}\stackrel{(7.17)}{\geqslant}\nu$. For this function, we have
$$
\begin{equation}
U(z)\equiv (u-h+M_s)(z)\underset{z\in \overline{\operatorname{str}}_b} {\stackrel{(7.16)}{\equiv}} M_{\mathbb{R}}(z)+M_s(z) \underset{z\in \overline{\operatorname{str}}_s} {\stackrel{(6.3)}{\equiv}} M(z)
\end{equation}
\tag{7.18}
$$
on the strip $\overline{\operatorname{str}}_b\subset \operatorname{str}_s$ outside some polar set. If two subharmonic functions coincide on an open set outside the polar set, these functions are equal everywhere on this open set. Thus, we constructed a subharmonic function $U$ of finite type such that $U(z)\equiv M(z)$ for all $z\in \operatorname{str}_b$, where $b$ is an arbitrarily number from the interval $(0,s)$ (see (7.10)). Consequently, it can be assumed that the identity $U(z)\equiv M(z)$ is satisfied for all $z$ from the closed strip $\overline{\operatorname{str}}_b$. This proves the implication II $\Rightarrow$ I of the main theorem. § 8. Lower bound of a subharmonic function by the logarithm of the modulus of the entire functionIf $u\not\equiv -\infty$ is an arbitrary subharmonic function on $\mathbb{C}$, then, of course, an entire function $f\not\equiv 0$ for which $\ln|f|\leqslant u$ on $\mathbb{C}$ or on a subset of $\mathbb{C}$ may fail to exist. One reason for this is that the $(-\infty)$-set of a subharmonic function $u$ may fail to be locally finite or may even be dense in $\mathbb{C}$. If some lower constraint of the form $\ln |f(z)|\leqslant u(z)$ is needed, two approaches are possible. The first one is to restrict to inequalities for points $z$ lying outside some (small, if possible) exceptional set $E\subset \mathbb{C}$. The second approach is to obtain lower bounds for integral averages of the form (2.10) or (2.7) over discs or circles of variable small radius $r\colon \mathbb{C}\to \mathbb{R}^+\setminus 0$. The latter option is often preferable because, first, it is additive, unlike the most commonly used lower bounds for the supremum of the function $u$ on discs $D_z\bigl(r(z)\bigr)$, and second, this approach is also capable of delivering lower bounds outside the exceptional sets $E$, as reflected in Theorem 2 of [50]. In this regard, we have the following result. Theorem 9. Let $u\not\equiv -\infty$ be a subharmonic function on $\mathbb{C}$, and let a function $r\colon \mathbb{C}\to (0,1]$ satisfy condition (2.17), which is equivalent to (2.24) in Remark 2. Then there exists an entire function $f\not\equiv 0$ such that Proof. Using [51], § 1.3, Corollary 2, with a comment after this corollary, which is based on [51], § 2.4, the proof of Corollary 2, see also assertion 4 of Lemma 5.1 in [52] and Theorem 1 in [50], we find that, for each subharmonic function $u\not\equiv -\infty$ on $\mathbb{C}$ and any number $P\in \mathbb{R}^+$ with particular function
$$
\begin{equation}
p\colon z\underset{z\in \mathbb{C}}{\longmapsto} \frac{1}{(1+|z|)^P},
\end{equation}
\tag{8.6}
$$
there exists an entire function $f_P\not\equiv 0$ such that $\ln|f_P(z)|\leqslant u^{\bullet p}(z)$ for all $z\in \mathbb{C}$. Hence, for the function $r$, with lower bounds (2.24) from Remark 2, we have $r(z)\geqslant p(z)$ and $\ln|f_P(z)|\leqslant u^{\bullet r}(z)$ for all $z\in \mathbb{C}\setminus R\overline{\mathbb{D}}$ with sufficiently large $R\in \mathbb{R}^+$. At the same time, by inequality (2.24) from Remark 2 we have $r(z)\geqslant c$ for all $z\in R\overline{\mathbb{D}}$ for some number $c\in \mathbb{R}^+\setminus 0$. Hence $u^{\bullet r}(z)\geqslant u^{\bullet c}(z)$ for all $z\in \overline{\mathbb{D}}$. As a result, for sufficiently small $a\in \mathbb{R}^+\setminus 0$, the entire function $f:=af_P$ satisfies the required estimate (8.2). Now (8.3) follows from the maximum principle.
To prove the final part of Theorem 9, for the functions $p\colon \mathbb C \to \mathbb R$, which are constructed or defined there, and which are, in general, different from the above concrete radial functions (8.6), we will take the supremum (2.9) over discs $D_z(s)$ of constant radius $s\in \mathbb R^+$, that is, Lemma 4. Let $\mu$ be a distribution of masses on $\mathbb{C}$, let $p\colon \mathbb{C} \to \mathbb{R}^+\setminus 0$ be a Borel bounded function, and let Then, for all $d\in (0,2]$ and $S\subset \mathbb{C}$, the set Proof. If, for a point $z\in \mathbb{C}$, we have
$$
\begin{equation}
\mu_z^{\mathrm{rad}}(t)\leqslant p^{-d}(z)t^d\quad\text{for all }t\leqslant p(z),
\end{equation}
\tag{8.11}
$$
then this point $z$ does not lie in the set $E$ (see (8.9)). Hence
$$
\begin{equation*}
\int_0^{p(z)}\frac{\mu_z^{\mathrm{rad}}(t)}{t}\,d t \stackrel{(8.11)}{\leqslant} p^{-d}(z)\int_0^{p(z)} t^{d-1}\,d t=\frac{1}{d}.
\end{equation*}
\notag
$$
Therefore, for each point $z\stackrel{(8.9)}{\in} E$, the negation of (8.11) gives and, clearly, the system of discs $\{\overline D_z(t_z)\}_{z\in E}$ covers the set $E$,
Theorem (the Besicovitch covering theorem; see Lemma 3.2 in [13], § 2.8.14 in [43], [53]–[55], § I.1, and the remarks in [56], and [57]). Let $t\colon z\underset{z\in E}{\longmapsto} t_z\in {\mathbb R}^+\setminus 0$ be a function on $E\subset\mathbb{C}$. If either $t$ or $E$ is bounded, then, for some $N\subset \mathbb{N}$, there exists a sequence of pairwise distinct points $z_k\in E$, $k\in N\subset \mathbb{N}$, such that $E\subset \bigcup_{k\in N} \overline D_{z_k}(t_{z_k})$ and the intersection of any $20$ different discs $\overline D_{z_k}(t_{z_k})$ is empty. By the Besicovitch covering theorem, we can replace (8.13) by an at most countable system of discs $\{\overline D_{z_k}(t_k)\}_{z_k\in E}$ which cover the set
$$
\begin{equation}
E\subset \bigcup_{z_k\in E} \overline D_{z_k}(t_k), \qquad 0<t_k:=t_{z_k}\leqslant p(z_k)\stackrel{(8.8)}{\leqslant} s.
\end{equation}
\tag{8.14}
$$
Consider an arbitrary $S\subset \mathbb{C}$. The set
$$
\begin{equation}
S_s:=\bigcup \, \{\overline D_{z_k}( t_k)\mid S\cap \overline D_{z_k}(t_k)\neq \varnothing\} \stackrel{(8.14)}{\subset} \bigcup_{z\in S}\overline D_z(2s) \stackrel{(2.8)}{=}S^{\cup (2s)}
\end{equation}
\tag{8.15}
$$
is, clearly, a Borel set. We also have the inequalities
$$
\begin{equation}
\begin{aligned} \, \sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} t_k^d &\stackrel{(8.12)}{\leqslant} \sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} p^d(z_k) \mu_{z_k}^{\mathrm{rad}}(t_k) \nonumber \\ &\,\,\,= \sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} \int_{\overline D_{z_k}(t_k)} p^d(z_k)\,d \mu(z). \end{aligned}
\end{equation}
\tag{8.16}
$$
Since $z_k\stackrel{(8.8)}{\in} \overline D_z(s)$ for all $z\in \overline D_{z_k}(t_k)$ and since $p^d(z_k)\stackrel{(8.8)}{\leqslant} (p^d)^{\vee s}(z)$ for all $z\in \overline D_{z_k}(t_k)$, we have by (8.16)
$$
\begin{equation*}
\begin{aligned} \, &\sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} t_k^d \stackrel{(8.7),(8.14)}{\leqslant} \sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} \int _{\overline D_{z_k}(t_k)}(p^d)^{\vee s} \,d \mu \\ &\stackrel{(8.15),(3.4)}{=} \!\! \sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} \int_{S_s} \boldsymbol{1}_{\overline D_{z_k}(t_k)} (p^d)^{\vee s} \,d \mu = \int_{S_s} \biggl(\sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} \!\boldsymbol{1}_{\overline D_{z_k}(t_k)}\biggr) (p^d)^{\vee s} \,d \mu. \end{aligned}
\end{equation*}
\notag
$$
Hence, by the Besicovitch covering theorem, the internal integrand in the last integral is majorized by $19$, and
$$
\begin{equation}
\sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} t_k^d\leqslant 19\int_{S_s} (p^d)^{\vee s} \,d \mu.
\end{equation}
\tag{8.17}
$$
In addition, by the definition of the $\mathfrak{m}_d^p$-Hausdorff content (2.4) of variable radius $p$, for each $\mu$-measurable set $S$, we have by (8.14), for $d\leqslant 2$,
$$
\begin{equation*}
\mathfrak{m}_d^p(E\cap S)\stackrel{(2.4)}{\leqslant} \dfrac{\pi^{d/2}}{{\Gamma(1+d}/2)} \sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} t_k^d\leqslant \pi\sum_{S\cap \overline D_{z_k}(t_k)\neq \varnothing} t_k^d.
\end{equation*}
\notag
$$
Now (8.10) is secured by inequality (8.17). Lemma 4 is proved. Lemma 5. Let $r\colon \mathbb{C}\to (0,1]$ be a function satisfying condition (2.17) (or the equivalent condition (2.24) from Remark 2), and $\mu$ be a distribution of masses of finite order on $\mathbb{C}$. Then, for any $d\in (0,2]$, there exists a function $p\colon \mathbb{C}\to\mathbb{R}^+\setminus 0$ which satisfies condition (2.17) and is such that Proof. Let $r$ be a function satisfying condition (2.17) in the form (2.24) from Remark 2. Then there exists a sufficiently large $Q\in \mathbb{R}^+$ such that
$$
\begin{equation}
\frac{1}{(2+|z|)^Q}\underset{z\in \mathbb{C}}{\leqslant}r(z).
\end{equation}
\tag{8.20}
$$
If $\operatorname{ord}[\mu]<l<+\infty$ as in the final part of Theorem 9, then, for $\mu$, there exists a number $C\geqslant 1$ such that
$$
\begin{equation}
\mu^{\mathrm{rad}}(t)\leqslant C(1+t)^l \quad\text{for all }t\in \mathbb{R}^+.
\end{equation}
\tag{8.21}
$$
For a given $d\in (0,2]$, we set and consider the function
$$
\begin{equation}
p\colon z\underset{z\in \mathbb{C}}{\longmapsto} \frac{1}{(60(l+1)C)^{1/d}(4+|z|)^P},
\end{equation}
\tag{8.23}
$$
for which, in view of $C\geqslant 1$, $d>0$ and $P\stackrel{(8.22)}{\geqslant} 1$, we have
$$
\begin{equation}
s\stackrel{(8.8)}{:=}\sup_{z\in \mathbb{C}} p(z)\leqslant \frac{1}{2}.
\end{equation}
\tag{8.24}
$$
Obviously, any function (8.23) satisfies condition (2.17), since
$$
\begin{equation}
\inf_{z\in \mathbb{C}}\frac{\ln p(z)}{\ln(2+|z|)}=-P>-\infty.
\end{equation}
\tag{8.25}
$$
In other words, such functions satisfy the first relation in (8.18). Therefore, we can further increase $P\geqslant 1$ unboundedly, remaining within the framework of condition (8.25) for the function $p$ from (8.23).
In notation (2.8), we have, by (8.24),
$$
\begin{equation}
S^{\cup(2s)}\stackrel{(8.24)}{\subset} S^{\cup 1} \quad\text{for any }S\subset \mathbb{C}.
\end{equation}
\tag{8.26}
$$
From (8.20), by the choice of $P$ in (8.22), and since $C\geqslant 1$, we have
$$
\begin{equation}
p(z)\stackrel{(8.23),(8.22)}{\underset{z\in \mathbb{C}}{\leqslant}} \frac{1}{(2+|z|)^Q} \underset{z\in \mathbb{C}} {\stackrel{(8.20)}{\leqslant}} r(z).
\end{equation}
\tag{8.27}
$$
This gives all the entire condition (8.18). By Lemma 4, for the set $E$, as defined in (8.9), and, for each subset $S\subset \mathbb{C}$, inequality (8.10) is satisfied, where
$$
\begin{equation}
S_s\subset S^{\cup(2s)}\stackrel{(8.26)}{\subset} S^{\cup 1}.
\end{equation}
\tag{8.28}
$$
In addition, for $d\in (0,2]$, in notation (8.7), we have
$$
\begin{equation}
(p^d)^{\vee s}(z) \stackrel{(8.24)}{\leqslant} (p^d)^{\vee \frac12}(z) \stackrel{(8.23),(8.22)}{\underset{z\in \mathbb{C}}{\leqslant}} \frac{1}{60C(l+1)(3+|z|)^{Q+l+1}}.
\end{equation}
\tag{8.29}
$$
Hence, for the integral with factor $60$ in the right-hand side of (8.10) with mass distribution of $\mu$, we have by (8.21)
$$
\begin{equation}
\begin{aligned} \, &60 \int_{S_s} (p^d)^{\vee s} \,d \mu \stackrel{(8.29)}{\leqslant} 60 \int_{S_s} \frac{\,d \mu(z)}{60C(l+1)(3+|z|)^{Q+l+1}} \nonumber \\ &\qquad\!\!\!\stackrel{(8.28)}{\leqslant} \sup_{z\in S^{\cup 1}} \frac{1}{(3+|z|)^Q} \int_{\mathbb{C}} \frac{d \mu(z)}{C(l+1)(3+|z|)^{l+1}} \nonumber \\ &\qquad= \frac{1}{(3+\inf_{z\in S} |z|-1)^Q} \int_0^{+\infty} \frac{d \mu^{\mathrm{rad}}(t)}{C(l+1)(3+t)^{l+1}} \nonumber \\ &\qquad\!\!\!\stackrel{(8.21)}{\leqslant} \sup_{z\in S} \frac{1}{(2+|z|)^Q} \int_0^{+\infty} \frac{C(1+t)^l\,d t}{C(3+t)^{l+2}} \leqslant \sup_{z\in S} \frac{1}{(2+|z|)^Q}\leqslant \sup_{z\in S} r(z). \end{aligned}
\end{equation}
\tag{8.30}
$$
Hence
$$
\begin{equation}
\mathfrak{m}_d^p(E\cap S) \stackrel{(8.10)}{\leqslant} 60 \int_{S_s} (p^d)^{\vee s} \,d \mu \stackrel{(8.30)}{\leqslant} \sup_{z\in S} r(z).
\end{equation}
\tag{8.31}
$$
But $r\geqslant p$ everywhere by the construction. Therefore, $\mathfrak{m}_d^r(E\cap S)\stackrel{(2.6)}{\leqslant} \mathfrak{m}_d^p(E\cap S)$. Now (8.19) follows from (8.31), proving Lemma 5. Let us return to the proof of the final part of Theorem 9 for the case of the Riesz distribution of masses $\mu:=\varDelta_u$ of finite order $\operatorname{ord}[\mu] =\operatorname{ord}[\varDelta_u]<+\infty$. Using Lemma 5, we can construct a function $p\colon \mathbb{C}\to (0,1]$ with properties (8.18), (8.19), and satisfying condition (2.17) (which is the first relation in (8.18)). Hence, by the already proved part of Theorem 9 with $p$ in place of $r$, there exists an entire function $f_p\not\equiv 0$ satisfying
$$
\begin{equation}
\ln |f_p(z)|\underset{z\in \mathbb{C}}{\stackrel{(8.2)}{\leqslant}} u^{\bullet p}(z) \underset{z\in \mathbb{C}} {\stackrel{(8.2)}{\leqslant}} u^{\circ p}(z).
\end{equation}
\tag{8.32}
$$
By the Poisson–Jensen–Privalov formula (see Ch. II, § 2 in [58], § 4.5 in [11], and § 3.7 in [12]), for the distribution of mass $\mu:=\varDelta_u$, we have
$$
\begin{equation*}
u^{\circ p}(z)= u(z)+\int_0^{p(z)}\frac{\mu^{\mathrm{rad}}(t)}{t}\,d t \quad\text{for all }z\stackrel{(8.1)}{\in} \mathbb C\setminus (-\infty)_u.
\end{equation*}
\notag
$$
Hence, using (8.32), we obtain
$$
\begin{equation}
\ln |f_p(z)|\stackrel{(8.32)}{\leqslant} u(z)+\frac1d \quad\text{for all } z\in \mathbb{C}\setminus \bigl(E\cup (-\infty)_u\bigr).
\end{equation}
\tag{8.33}
$$
By definition (8.9) of the set $E$ in Lemma 5, where $(-\infty)_u\subset E$ and $E\,{\cup}\, (-\infty)_u= E$, we have
$$
\begin{equation*}
\int_0^{p(z)}\frac{\mu^{\mathrm{rad}}(t)}{t}\,d t=+\infty \quad\text{for all }z\in (-\infty)_u.
\end{equation*}
\notag
$$
We set $f:=f_pe^{-1/d}\not\equiv 0$. By (8.33), we have (8.4), that is,
$$
\begin{equation}
\ln |f(z)|\underset{z\in \mathbb{C}}{\equiv}\ln |f_p(z)|-\frac{1}{d} \stackrel{(8.33)}{\leqslant} u(z)\quad\text{for all } z\in \mathbb{C}\setminus E,
\end{equation}
\tag{8.34}
$$
where by Lemma 5, the set $E$ satisfies (8.19) (that is, (8.5)) for each subset $S\subset \mathbb{C}$. At the same time, by (8.32) and by the inequality $p\stackrel{(8.18)}{\leqslant} r$ on $\mathbb{C}$, the function $\ln|f|=\ln|f_p|-1/d$ still satisfies (8.2):
$$
\begin{equation*}
\ln |f(z)|\stackrel{(8.34)}{\underset{z\in \mathbb{C}}{\leqslant}} \ln |f_p(z)|\underset{z\in \mathbb{C}}{\stackrel{(8.32)}{\leqslant}} u^{\bullet p}(z) \underset{z\in \mathbb{C}}{\stackrel{(8.18)}{\leqslant}} u^{\bullet r}(z)\underset{z\in \mathbb{C}}{\stackrel{(2.11)}{\leqslant}} u^{\circ r}(z)\underset{z\in \mathbb{C}}{\leqslant} u^{\vee r}(z).
\end{equation*}
\notag
$$
Now (8.3) follows from the maximum principle. This completes the proof of Theorem 9.
§ 9. Proofs of the implications I $\Rightarrow$ III $\Rightarrow$ IV of the main theorem Proof of the implication I $\Rightarrow$ III. An application assertion I with arbitrary $b'\in (b,s)$ in place of $b\in [0,s)$ shows that there exists a subharmonic function $U\not\equiv -\infty$ of finite type with Riesz distribution of masses $\varDelta_U\geqslant \nu$ satisfying
$$
\begin{equation}
U(z)\stackrel{(2.14)}{ \underset{z\in \overline{\operatorname{str}}_{b'}}{\equiv}} M(z).
\end{equation}
\tag{9.1}
$$
The function $U$ can be represented as the sum of three subharmonic functions
$$
\begin{equation}
U=v+m+u, \qquad v\not\equiv -\infty, \qquad m\not\equiv -\infty, \qquad u\not\equiv -\infty,
\end{equation}
\tag{9.2}
$$
with Riesz distributions masses of finite upper density, respectively
$$
\begin{equation}
\varDelta_v=\nu, \qquad \varDelta_m= \varDelta_M\lfloor_{\operatorname{str}_{s}}, \qquad \varDelta_u=\varDelta_U-\varDelta_M\lfloor_{\operatorname{str}_{s}}-\nu.
\end{equation}
\tag{9.3}
$$
By identity (9.1), $\varDelta_U\lfloor_{\operatorname{str}_{b'}} = \varDelta_M\lfloor_{\operatorname{str}_{b'}}$, and $ \operatorname{str}_{b'}\cap \operatorname{supp} \nu =\varnothing$ in view of (2.13). Hence $\operatorname{str}_{b'} \cap \operatorname{supp}\varDelta_u\stackrel{(9.3)}{=}\varnothing$, and the subharmonic function $u$ is harmonic on an open vertical strip $\operatorname{str}_{b'}$ of width $2b'>2b$. The constant function
$$
\begin{equation}
r\colon z\underset{z\in \mathbb{C}}{\longmapsto} \min\biggl\{\frac12(b'-b),\, 1\biggr\}>0,
\end{equation}
\tag{9.4}
$$
obviously satisfies condition (2.17) of Theorem 9. By this result, there exists an entire function $h\not\equiv 0$ such that, for the function $r$ from (9.4), we have
$$
\begin{equation}
\ln|h(z)|\stackrel{(8.2)}{\leqslant} u^{\circ r}(z) \quad\text{for all }z\in \mathbb{C}.
\end{equation}
\tag{9.5}
$$
From here, using representation (9.2) on $\mathbb{C}$, we obtain the inequality which can be continued as where the function $U^{\circ 1}$ on the right is subharmonic and is of finite type. Consequently, so is the function $v+m+\ln|h|$.
In addition, by the choice of the constant function $r$ (see (9.4)), for the function $u$, which is harmonic on the strip $\operatorname{str}_{b'}$ of width $2b'>2b$, we have the identity
$$
\begin{equation}
u^{\circ r}(z)\underset{z\in \overline{\operatorname{str}}_b}{\equiv}u(z).
\end{equation}
\tag{9.7}
$$
Hence by (9.6) we have
$$
\begin{equation*}
v(z)+m(z)+\ln|h(z)|\stackrel{(9.6)}{\leqslant} v(z)+m(z)+u^{\circ r}(z) \underset{z\in \overline{\operatorname{str}}_b} {\stackrel{(9.7)}{\equiv}} v(z)+m(z)+u(z).
\end{equation*}
\notag
$$
By (9.2), the right-hand side in the above formila is identically $U(z)$ for all $z\in \mathbb{C}$, and identity (9.1) secures (2.16), which gives assertion III. Proof of the implication III $\Rightarrow$ IV. For $b\in [0,s)$, we choose $b'\in (b,s)$ and replace the function $r$ from (2.17) by the smaller function
$$
\begin{equation}
r_*(z)\underset{z\in \mathbb{C}}{:=} \min\biggl\{r(z),\, \frac12(b'-b)\biggr\}\leqslant r(z)\leqslant 1,
\end{equation}
\tag{9.8}
$$
for which, obviously, condition (2.17) is still satisfied, and
Let assertion III hold with $b'$ in place of $b$. An application of Theorem 9 with the function $r_*$ instead of $r$ to the subharmonic function $m$ with Riesz distribution of masses $\frac{1}{2\pi}\Delta m=\frac{1}{2\pi} \Delta M\lfloor_{\operatorname{str}_{s}}$ of finite upper density gives us an entire function $f\not\equiv 0$ such that
$$
\begin{equation}
\ln |f(z)| \stackrel{(2.10)}{\leqslant} m^{\bullet r_*}(z) \quad \text{for all }z\in \mathbb{C},
\end{equation}
\tag{9.10}
$$
$$
\begin{equation}
\ln |f(z)| \,\,\,\,\leqslant m(z) \quad \text{for all } z\in \mathbb{C}\setminus E.
\end{equation}
\tag{9.11}
$$
Here, for the set $E\subset \mathbb{C}$, we have the inequality
$$
\begin{equation}
\mathfrak{m}_d^{r_*}(E\cap S)\stackrel{(8.5)}{\leqslant} \sup_{z\in S} r_*(z) \quad \text{for each } S\subset \mathbb{C}.
\end{equation}
\tag{9.12}
$$
But from the inequality $r(z)\underset{z\in \mathbb{C}}{\stackrel{(9.8)}{\geqslant}} r_*(z)$ we have
$$
\begin{equation*}
\mathfrak{m}_d^r(E\cap S)\stackrel{(2.6)}{\leqslant} \mathfrak{m}_d^{r_*}(E\cap S)\stackrel{(9.12)}{\leqslant} \sup_{z\in S} r_*(z)\leqslant \sup_{z\in S} r(z) \quad\text{for any }S\subset \mathbb{C},
\end{equation*}
\notag
$$
which gives (2.20) for $E_b:=E$. In addition, applying (2.16) and (9.11), we have
$$
\begin{equation*}
v(z)+\ln |f(z)h(z)|\underset{z\in \mathbb{C}} {\stackrel{(9.11)}{\leqslant}} v(z)+m(z)+ \ln|h(z)|\stackrel{(2.16)}{\leqslant} M(z)\quad\text{for all } z\in \overline{\operatorname{str}}_{b'}\setminus E_b,
\end{equation*}
\notag
$$
which gives (2.19), (2.20) from assertions IV with $h\not\equiv 0$ in place of the of entire function $fh\not\equiv 0$.
Applying the integral averages (2.10) over the discs $D_z(r(z))$ to inequality (2.16) in assertion III, we obtain
$$
\begin{equation*}
\begin{aligned} \, v(z)+\ln |f(z)|+\ln|h(z)| &\stackrel{(9.10)}{\leqslant} v(z)+m^{\bullet r_*}(z)+(\ln|h|)(z) \\ &\stackrel{(2.11)}{\leqslant} v^{\bullet r_*}(z)+m^{\bullet r_*}(z)+ (\ln|h|)^{\bullet r_*}(z) \\ &\stackrel{(2.10)}{=} (v+m+\ln|h|)^{\bullet r_*}(z) \quad\text{for all } z\in \mathbb{C}. \end{aligned}
\end{equation*}
\notag
$$
Hence $v+\ln |fh|$ is a subharmonic function of finite type, since by assertion III so is the function $v+m+\ln|h|$ and the function $r_*$ is bounded. In addition, from the extreme parts of these inequalities and using (2.16) and (9.9), we have
$$
\begin{equation*}
v(z)+\ln|f(z)h(z)|\leqslant (v+m+\ln|h|)^{\bullet r_*}(z) \stackrel{(2.16),(9.9)}{\leqslant} M^{\bullet r_*}(z) \quad\text{for all } z\in \overline{\operatorname{str}}_b.
\end{equation*}
\notag
$$
Hence, for the entire function $fh\not\equiv 0$,
$$
\begin{equation*}
v(z)+\ln |(fh)(z)|\stackrel{(2.16),(9.9)}{\leqslant} M^{\bullet r_*}(z)\stackrel{(9.8)}{\leqslant} M^{\bullet r}(z) \quad\text{for all } z\in \overline{\operatorname{str}}_b.
\end{equation*}
\notag
$$
Denoting the entire function $fh$ by $h$, we arrive at (2.18). This proves the implication III $\Rightarrow$ IV, and therefore, the main theorem. § 10. Complementing the distribution of masses to satisfy the Lindelöf conditionsIn this section, we consider a distribution of masses $\mu$ of finite upper density satisfying the following condition:
Condition [$ \boldsymbol\mu ^{\operatorname{rh}}$] with (10.1) is met under either of the following five conditions:
Here, obviously, [$ \boldsymbol\mu 1$] follows from [$ \boldsymbol\mu 2$], [$ \boldsymbol\mu 3$] follows from [$ \boldsymbol\mu 4$], and [$ \boldsymbol\mu 4$] is secured by [$ \boldsymbol\mu 5$] by the Weierstrass–Hadamard–Lindelöf–Brelot theorem. Given $S\subset \mathbb{C}$, let $\overline{S}:=\{\overline z\mid z\in S\}$ be the set reflection symmetric to $S$ about the real axis $\mathbb{R}$. If $\nu$ is a distribution of charges on $\mathbb{C}$, then the charge distribution that is reflection symmetric to it about to the imaginary axis $i\mathbb{R}$ is defined by
$$
\begin{equation}
\nu^{\leftrightarrow}(S):=\nu(-\overline S)\quad \text{for all Borel subsets }S\subset \mathbb{C}.
\end{equation}
\tag{10.2}
$$
Proposition 11. Let $\mu$ be a distribution of masses of finite upper density with property [$ \boldsymbol\mu ^{\operatorname{rh}}$]. Then there exists a distribution of masses $\gamma$ of finite upper density, $\operatorname{supp} \gamma\subset \mathbb{R}^+$, such that $\mu+\gamma$ satisfies the Lindelöf $\mathbb{R}$-condition (3.18), (3.19), and Proof. Without loss of generality we can assume that $0\notin \operatorname{supp} \mu$. Let us represent the mass distribution $\mu$ as the sum of its two restrictions
$$
\begin{equation}
\mu=\mu_{\overline{\mathrm{rh}}}+\mu_{\mathrm{lh}}, \qquad \mu_{\overline{\mathrm{rh}}}:= \mu\lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}, \qquad \mu_{\mathrm{lh}}:=\mu\lfloor_{\mathbb{C}_{\mathrm{lh}}},
\end{equation}
\tag{10.4}
$$
to the closed right and open left half-planes, respectively. For the mass distribution $\mu_{\mathrm{lh}}$, let $\mu_{\mathrm{lh}}^{\leftrightarrow}$ (see (10.2)) be the mass distribution reflection symmetric to $\mu$ about $i\mathbb{R}$. By construction,
$$
\begin{equation}
0\notin \operatorname{supp} \mu_{\overline{\mathrm{rh}}}\cup \operatorname{supp} \mu_{\mathrm{lh}}^{\leftrightarrow}\subset \mathbb{C}_{\overline{\mathrm{rh}}}.
\end{equation}
\tag{10.5}
$$
For any $0<r<R<+\infty$, by definitions (2.1), (2.2), we have
$$
\begin{equation*}
\begin{aligned} \, \ell_{\mu}^{\operatorname{lh}}(r,R)-\ell_{\mu}^{\operatorname{rh}}(r,R) &\stackrel{(10.4)}{=}\ell_{\mu_{\mathrm{lh}}}^{\operatorname{lh}}(r,R) -\ell_{\mu_{\overline{\mathrm{rh}}}}^{\operatorname{rh}}(r,R) \\ &\stackrel{(10.2)}{=} \ell_{\mu_{\mathrm{lh}}^{\leftrightarrow}}^{\operatorname{rh}}(r,R) -\ell_{\mu_{\overline{\mathrm{rh}}}}^{\operatorname{rh}}(r,R) \stackrel{(10.5)}{=} \ell_{\mu_{\mathrm{lh}}^{\leftrightarrow}}(r,R)- \ell_{\mu_{\overline{\mathrm{rh}}}}(r,R). \end{aligned}
\end{equation*}
\notag
$$
Hence by condition (10.1)
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_{\mu_{\mathrm{lh}}^{\leftrightarrow}}(r_n,r_N) -\ell_{\mu_{\overline{\mathrm{rh}}}}(r_n,r_N)\bigr)<+\infty,
\end{equation}
\tag{10.6}
$$
and further, using the implication (7.2) $\Rightarrow$ (7.3) in Proposition 10, we have, by (7.1) and (10.5),
$$
\begin{equation}
\sup_{0< r<R<+\infty} \bigl(\ell_{\mu_{\mathrm{lh}}^{\leftrightarrow}}(r,R) -\ell_{\mu_{\overline{\mathrm{rh}}}}(r,R)\bigr)<+\infty.
\end{equation}
\tag{10.7}
$$
Consider the distribution of charges
$$
\begin{equation}
\eta:=\mu_{\mathrm{lh}}^{\leftrightarrow}-\mu_{\overline{\mathrm{rh}}},\qquad \operatorname{supp} \eta\stackrel{(10.5)}{\subset} \mathbb{C}_{\overline{\mathrm{rh}}},
\end{equation}
\tag{10.8}
$$
of finite upper density. For the distribution, (10.7) means that
$$
\begin{equation}
\sup_{0< r<R<+\infty} \ell_{\eta}^{\operatorname{rh}}(r,R)<+\infty.
\end{equation}
\tag{10.9}
$$
By the construction used in Proposition 4, there exists a distribution of masses $\alpha$ of finite upper density with $\operatorname{supp} \alpha\subset \mathbb{R}^+\setminus 0$ such that
$$
\begin{equation*}
\sup_{0\leqslant r<R<+\infty} |\ell_{\eta+\alpha}^{\operatorname{rh}}(r,R)| \stackrel{(4.4)}{\leqslant} 2 \sup_{0\leqslant r<R<+\infty} \ell_{\eta}^{\operatorname{rh}}(r,R)\stackrel{(10.9)}{<}+\infty.
\end{equation*}
\notag
$$
According to (10.8), these inequalities can be written as
$$
\begin{equation}
\sup_{1\leqslant r<R<+\infty} \bigl|\ell_{\mu_{\mathrm{lh}}^{\leftrightarrow}+ \alpha}^{\operatorname{rh}}(r,R) -\ell_{\mu_{\overline{\mathrm{rh}}}}^{\operatorname{rh}}(r,R)\bigr|<+\infty.
\end{equation}
\tag{10.10}
$$
Setting $\gamma\stackrel{(10.2)}{:=}\alpha^{\leftrightarrow}$, we have $\operatorname{supp} \gamma \subset -\mathbb{R}^+$, inasmuch as $\operatorname{supp} \alpha\subset \mathbb{R}^+$, and since
$$
\begin{equation}
\begin{aligned} \, \ell_{\mu+\gamma}^{\operatorname{lh}}(r,R)- \ell_{\mu+\gamma}^{\operatorname{rh}}(r,R) &=\ell_{\mu+\gamma}^{\operatorname{lh}}(r,R)- \ell_{\mu}^{\operatorname{rh}}(r,R) \nonumber \\ &=\ell_{\mu_{\mathrm{lh}}^{\leftrightarrow}+ \gamma^{\leftrightarrow}}^{\operatorname{rh}}(r,R) -\ell_{\mu}^{\operatorname{rh}}(r,R) =\ell_{\mu_{\mathrm{lh}}^{\leftrightarrow}+\alpha}^{\operatorname{rh}}(r,R) -\ell_{\mu}^{\operatorname{rh}}(r,R). \end{aligned}
\end{equation}
\tag{10.11}
$$
Hence, since $\ell_{\mu+\gamma}^{\operatorname{rh}}(r,R) =\ell_{\mu}^{\operatorname{rh}}(r,R)\leqslant \ell_{\mu}(r,R)$, and in view of (2.1)–(2.3),
$$
\begin{equation*}
\begin{aligned} \, &\sup_{1\leqslant r<R<+\infty} \bigl(\ell_{\mu+\gamma}(r,R)-\ell_{\mu}(r,R) \bigr) \leqslant \sup_{1\leqslant r<R<+\infty} \bigl(\ell_{\mu+\gamma}(r,R) -\ell_{\mu}^{\operatorname{rh}}(r,R) \bigr) \\ &\qquad\stackrel{(2.3)}{=}\sup_{1\leqslant r<R<+\infty} \bigl(\ell_{\mu+\gamma}^{\operatorname{lh}}(r,R)- \ell_{\mu}^{\operatorname{rh}}(r,R) \bigr)^+ \\ &\qquad\!\!\stackrel{(10.11)}{\leqslant} \sup_{1\leqslant r<R<+\infty} \bigl|\ell_{\mu_{\mathrm{lh}}^{\leftrightarrow}+ \alpha}^{\operatorname{rh}}(r,R) -\ell_{\mu_{\overline{\mathrm{rh}}}}^{\operatorname{rh}}(r,R)\bigr| \stackrel{(10.10)}{<}+\infty. \end{aligned}
\end{equation*}
\notag
$$
Consequently, (10.3) holds by these inequalities and by the obvious inequality $\ell_{\mu+\gamma}(r,R)\geqslant\ell_{\mu}(r,R)$. In addition, from (10.11) we have
$$
\begin{equation*}
\begin{aligned} \, &\sup_{1\leqslant r<R<+\infty} \|\ell_{\mu+\gamma}^{\operatorname{lh}}(r,R) -\ell_{\mu+\gamma}^{\operatorname{rh}}(r,R)| \\ &\qquad=\sup_{1\leqslant r<R<+\infty} \bigl|\ell_{\mu_{\mathrm{lh}}^{\leftrightarrow}+ \alpha}^{\operatorname{rh}}(r,R)-\ell_{\mu}^{\operatorname{rh}}(r,R)\bigr| \stackrel{(10.10)}{<}+\infty. \end{aligned}
\end{equation*}
\notag
$$
Thus, the distribution of masses $\mu+\gamma$ satisfies the Lindelöf $\mathbb{R}$-condition (3.19). This proves Proposition 11. Proposition 12. If $\nu$ is a distribution of masses of finite upper density, then there exists a mass distribution $\beta$ of finite upper density such that $\operatorname{supp} \beta\subset i\mathbb{R}$ and $\nu+\beta$ satisfies the Lindelöf $i\mathbb{R}$-condition (3.20). Proof. Without loss of generality we can assume that $0\notin \operatorname{supp} \nu$. Let be the clockwise rotation through a right angle of the distribution of masses $\nu$.
Let us represent the distribution of masses $\nu^{\circlearrowright}$ as the sum of its two restrictions
$$
\begin{equation}
\nu^{\circlearrowright}=\nu^{\circlearrowright}_{\mathrm{rh}} + \nu^{\circlearrowright}_{\mathrm{lh}},\qquad \nu^{\circlearrowright}_{\mathrm{rh}} :=\nu^{\circlearrowright} \lfloor_{\mathbb{C}_{\overline{\mathrm{rh}}}}, \qquad \nu^{\circlearrowright}_{\mathrm{lh}}:=\mu\lfloor_{\mathbb{C}_{\mathrm{lh}}},
\end{equation}
\tag{10.13}
$$
to the closed right and open left half-planes, respectively. With the left-hand component $\nu^{\circlearrowright}_{\mathrm{lh}}$ we associate some distribution of masses $\gamma_{\mathrm{rh}}$ of finite upper density concentrated on the positive semi-axis and such that
$$
\begin{equation}
\ell_{\nu^{\circlearrowright}}^{\operatorname{lh}}(2^n,2^{n+1}) \stackrel{(2.2)}{=}\ell_{\nu^{\circlearrowright}_{\mathrm{lh}}} (2^n,2^{n+1})\leqslant \ell_{\gamma_{\mathrm{rh}}}(2^n,2^{n+1}) \quad\text{for all }n\in \mathbb{N}_0.
\end{equation}
\tag{10.14}
$$
This can be done, for example, by employing the following interval-point-by-point method. For each $n\in \mathbb{N}_0$ and the corresponding interval $(2^n,2^{n+1}]$, we place at the right end of $2^{n+1}$ of this interval a mass equal to
$$
\begin{equation}
g_n:=2^{n+1}\ell_{\nu^{\circlearrowright}_{\mathrm{lh}}}^{\operatorname{lh}} (2^n,2^{n+1}) \stackrel{(2.2)}{\leqslant} 2^{n+1}\frac{1}{2^n} \nu^{\mathrm{rad}}(2^{n+1}) \leqslant 2\nu^{\mathrm{rad}}(2^{n+1}),
\end{equation}
\tag{10.15}
$$
and, for the distribution of masses $\gamma_{\mathrm{rh}}$, we take the sum of all these masses $g_n$ concentrated at the points $2^{n+1}$. By construction and from (10.15) it follows that
$$
\begin{equation*}
\gamma_{\mathrm{rh}}^{\mathrm{rad}}(2^{n+1})- \gamma_{\mathrm{rh}}^{\mathrm{rad}}(2^{n})=g_n\leqslant 2\nu^{\mathrm{rad}}(2^{n+1})\quad\text{for all }n\in \mathbb{N},
\end{equation*}
\notag
$$
whence $\gamma_{\mathrm{rh}}$ is the mass distribution of finite upper density, and $\operatorname{supp} \gamma_{\mathrm{rh}} \subset \mathbb{R}^+$. In addition, by construction (10.15) of masses $g_n$ at points $2^{n+1}$, we have
$$
\begin{equation*}
\ell_{\gamma_{\mathrm{rh}}}^{\operatorname{rh}}(2^n,2^{n+1}) \stackrel{(2.1)}{=} \frac{1}{2^{n+1}}g_n\stackrel{(10.15)}{=} \ell_{\nu^{\circlearrowright}_{\mathrm{lh}}}^{\operatorname{lh}}(2^n,2^{n+1}).
\end{equation*}
\notag
$$
In particular, (10.14) is fulfilled. Consider the distribution of masses
$$
\begin{equation}
\mu:=\nu^\circlearrowright+\gamma_{\mathrm{rh}}, \qquad \mu\lfloor_{\mathbb{C}_{\mathrm{lh}}}=\nu^{\circlearrowright}_{\mathrm{lh}}, \qquad \mu\lfloor_{\mathbb{C}_{\mathrm{rh}}}= \nu^{\circlearrowright}_{\mathrm{lh}}+\gamma_{\mathrm{rh}}
\end{equation}
\tag{10.16}
$$
of finite upper density. From (10.14) it follows that this distribution satisfies condition [$ \boldsymbol\mu ^{\operatorname{rh}}$] with (10.1) for the sequence of numbers $r_n\underset{n\in \mathbb{N}_0}{:=}2^n$. By Proposition 11, for the distribution of masses $\mu$, there exists a distribution of masses $\gamma_{\mathrm{lh}}$ of finite upper density, $\operatorname{supp} \gamma\subset -\mathbb{R}^+$, such that $\mu+\gamma_{\mathrm{lh}}$ satisfies the Lindelöf $\mathbb{R}$-condition (3.18), (3.19). If we set $\gamma:=\gamma_{\mathrm{lh}}+\gamma_{\mathrm{rh}}$, then by construction $\gamma$ is the distribution of masses of finite upper density with $\operatorname{supp} \gamma\subset \mathbb{R}$, and $\nu^\circlearrowright+\gamma\stackrel{(10.16)}{=} \mu+\gamma_{\mathrm{lh}}$ satisfies the Lindelöf $\mathbb{R}$-condition (3.18), (3.19). Obviously, there exists a distribution of masses $\beta$ of finite upper density, $\operatorname{supp} \beta\subset i\mathbb{R}$, such that its clockwise rotation through a right angle gives $\gamma\stackrel{(10.12)}{=}\beta^{\circlearrowright}$. Hence
$$
\begin{equation*}
(\nu+\beta)^\circlearrowright\stackrel{(10.12)}{=} \nu^\circlearrowright+\beta^\circlearrowright= \nu^\circlearrowright+\gamma
\end{equation*}
\notag
$$
is the distribution of masses of finite upper density that satisfies the Lindelöf $\mathbb{R}$-condition (3.18), (3.19). Accordingly, by definition (3.20), the distribution of masses $\nu+\beta$ of finite upper density satisfies the Lindelöf $i\mathbb{R}$-condition (3.20). This completes the proof of Proposition 12. Proposition 13. If $\mu$ is a distribution of masses of finite upper density with property [$ \boldsymbol\mu ^{\operatorname{rh}}$], then there exists a distribution of masses $\varDelta\geqslant \mu$ of finite upper density satisfying the Lindelöf condition (3.21) and such that Proof. By Proposition 11, there exists a distribution of masses $\gamma$ of finite upper density such that $\operatorname{supp} \gamma\subset -\mathbb{R}^+$, $\mu+\gamma$ satisfies the Lindelöf $\mathbb{R}$-condition (3.18), (3.19), and (10.3) is met. By Proposition 12, for the distribution of masses $\nu:=\mu+\gamma$, there exists a distribution of masses $\beta$ of finite upper density such that $\operatorname{supp} \beta\subset i\mathbb{R}$, and $\nu+\beta=\mu+\gamma+\beta$ satisfies the Lindelöf $i\mathbb{R}$-condition (3.20).
We set $\varDelta:=\mu+\gamma+\beta$. Since $\operatorname{supp} \gamma \cup \operatorname{supp} \beta\subset (-\mathbb{R}^+)\cup i\mathbb{R}$, we obtain (10.17), and now (10.18) follows from the equality $\ell_{\varDelta}=\ell_{\mu+\gamma}$ and (10.3). This completes the proof of Proposition 13. Remark 10. Instead of condition [$ \boldsymbol\mu ^{\operatorname{rh}}$], we can consider the condition which is reflection symmetric about the imaginary axis: Remark 11. Theorem 3.2 in the recent paper [38] of the author of the present paper and Salimova is concerned with the complement of the distribution of points from $\mathrm{Z}\subset \mathbb{C}$ of finite upper density to that of points of finite upper density satisfying the Lindelöf condition. In the proof of this result, the sequence $(r_n)_{n\in \mathbb{N}}$ from (7.1) was implicitly assumed to satisfy the condition § 11. Variants of the main results for pairs of distributions of masses or points11.1. Developments of the Malliavin–Rubel theoremThe Malliavin–Rubel theorem and Theorem 3 formulated in § 1.3 are the starting points for the present study. They consider the distributions of points $\mathrm{Z}\subset \mathbb{C}$ and $\mathrm{W}\subset \mathbb{C}_{\mathrm{rh}}$ of finite upper density. Developing this statement further, we will consider, in this section, the distribution of masses $\mu$ instead of that of points $\mathrm{W}\subset \mathbb{C}_{\mathrm{rh}}$. Theorem 10. Let $\mu$ be a distribution of masses of finite upper density with property [$ \boldsymbol\mu ^{\operatorname{rh}}$]. Suppose that a distribution of masses $\nu$ satisfy conditions (2.13). Then the following five assertions I– V are equivalent. I. For any $b\in [0,s)$ and each subharmonic function $M\not\equiv -\infty$ of finite type with Riesz distribution of masses $\frac{1}{2\pi}\Delta M\geqslant\mu$, there exists a subharmonic function $U\not\equiv -\infty$ of finite type with Riesz distribution of mass $\frac{1}{2\pi}\Delta U\geqslant \nu$ satisfying (2.14). II. If $n\in \mathbb{N}_0$ and $N\in \mathbb{N}$, then
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_{\nu}(2^n,2^N) -\ell_{\mu}(2^n,2^N)\bigr)<+\infty.
\end{equation}
\tag{11.1}
$$
III. For each subharmonic function $M\not\equiv -\infty$ of finite type with Riesz distribution of masses $\frac{1}{2\pi}\Delta M\geqslant\mu$ and each pair of subharmonic functions $v$ and $m$ with Riesz distributions of masses, respectively $\frac{1}{2\pi}\Delta v=\nu$ and $\frac{1}{2\pi}\Delta m=\frac{1}{2\pi}\Delta M \lfloor_{\operatorname{str}_{s}}$, for any $b\in [0,s)$, there exists an entire function $h\not\equiv 0$ such that the sum $v+m+\ln |h|$ is a subharmonic function of finite type and inequalities (2.16) hold. IV. For each subharmonic function $M\not\equiv -\infty$ of finite type with Riesz distribution of masses $\frac{1}{2\pi}\Delta M\geqslant\mu$, and, for an arbitrary subharmonic function $v$ with Riesz distribution of masses $\frac{1}{2\pi}\Delta v=\nu$, for all $b\in [0,s)$, $d\in (0,2]$, and any function $r\colon \mathbb{C}\to (0,1]$ satisfying (2.17), there exist an entire function $h\not\equiv 0$ and a subset $E_b\subset \mathbb{C}$ such that $v+\ln |h|$ is a subharmonic function of finite type and (2.18)–(2.20) hold. IV. There exist a subharmonic function $M\not\equiv -\infty$ of finite type satisfying
$$
\begin{equation}
\varDelta_M\stackrel{(1.12)}{:=}\frac{1}{2\pi}\Delta M\geqslant\mu, \qquad \varDelta_M\lfloor_{\mathbb{C}_{\mathrm{rh}}}= \mu\lfloor_{\mathbb{C}_{\mathrm{rh}}},
\end{equation}
\tag{11.2}
$$
and functions $q_0$, $q$, a subset $E\subset \mathbb{C}$, and a subharmonic function $U\not\equiv -\infty$ with the same properties as in assertion V of the main theorem (see (2.21) and (2.22)). Proof. It is easily checked that the implications I $\Rightarrow$ III $\Rightarrow$ IV are precisely the implications I $\Rightarrow$ III $\Rightarrow$ IV of the main theorem. Let us prove the implications IV $\Rightarrow$ V $\Rightarrow$ II $\Rightarrow$ I.
Let us show that assertion V follows from assertion IV. By Proposition 13, there exists a distribution of masses $\varDelta\geqslant \mu$ of finite upper density satisfying the Lindelöf condition (3.21) with properties (10.17)–(10.18). By the Weierstrass–Hadamard–Lindelöf–Brelot theorem, there exists a subharmonic function $M\not\equiv -\infty$ with Riesz distribution of masses $\frac{1}{2\pi}\Delta M=\varDelta\geqslant \mu$. By assertion IV, for $b:=0$, there exists a subharmonic function of finite type $U:=v+\ln|h|\not\equiv -\infty$ from (2.18)–(2.20) with Riesz distribution of masses $\frac{1}{2\pi}\Delta U\geqslant \nu$ satisfying the inequality $U(iy)\leqslant M(iy)$ for all $iy\in (\mathbb{C}\setminus E_0)\cap i\mathbb{R}$, where, for $d:=1$ and with $r\equiv 1$ in (2.17), we have $\mathfrak{m}_1^1(E_0) <+\infty$. Now from assertion IV we have $U(iy)+U(-iy)\leqslant M(iy)+M(-iy)$ for all $y\in \mathbb{R}^+\setminus E$, where $E:=((iE_0)\cup (-iE_0))\cap \mathbb{R}^+$ of finite Lebesgue measure $\mathfrak{m}_1(E) <+\infty$. For this set $E\subset \mathbb{R}^+$ we have (2.37). With $q_0=q=0$ it follows that the integral in (2.22) in finite and inequalities (2.21) hold for all $y\in \mathbb{R}^+\setminus E$. This proves the implication IV $\Rightarrow$ V. If assertion V holds, then (2.15) follows from the implication V $\Rightarrow$ II of the main theorem. Hence, by using (6.10) of Lemma 3 in the form (6.7), we get
$$
\begin{equation}
\begin{aligned} \, &\limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_{\nu}(2^n,2^N)- \ell_{\varDelta_M}(2^n,2^N)\bigr) \nonumber \\ &\qquad\leqslant \limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_{\nu}(2^n,2^N) -J_{i\mathbb{R}}(2^n,2^N;M)\bigr) \nonumber \\ &\qquad\qquad+\limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl|J_{i\mathbb{R}}(2^n,2^N;M)-\ell_{\varDelta_M}(2^n,2^N)\bigr| \stackrel{(2.15),(6.10)}{<}+\infty. \end{aligned}
\end{equation}
\tag{11.3}
$$
The distribution of masses $\varDelta_M$ satisfies the Lindelöf condition, and, a fortiori, the Lindelöf $\mathbb{R}$-condition in the form (3.19). Hence
$$
\begin{equation*}
\sup_{1\leqslant r<R<+\infty}\bigl|\ell_{\varDelta_M}(r,R) -\ell_{\varDelta_M}^{\operatorname{rh}}(r,R)\bigr|<+\infty.
\end{equation*}
\notag
$$
Now by (11.3) we obtain
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_{\nu}(2^n,2^N) -\ell_{\varDelta_M}^{\operatorname{rh}}(2^n,2^N)\bigr) <+\infty.
\end{equation}
\tag{11.4}
$$
In view of $\varDelta_M\lfloor_{\mathbb{C}_{\mathrm{rh}}} \stackrel{(11.2)}{=} \mu\lfloor_{\mathbb{C}_{\mathrm{rh}}}$, we have $\ell_{\varDelta_M}^{\operatorname{rh}} =\ell_{{\varDelta_M}\lfloor_{\mathbb{C}_{\mathrm{rh}}}} =\ell_{\mu\lfloor_{\mathbb{C}_{\mathrm{rh}}}} =\ell_{\mu}^{\operatorname{rh}}\leqslant \ell_{\mu}$. Now using (11.4) we obtain (11.1) and assertion II.
From (11.1) in assertion II it follows that, for every subharmonic function $M\not\equiv -\infty$ of finite type with Riesz distribution of masses $\varDelta_M=\frac{1}{2\pi}\Delta M\geqslant\mu$,
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n<N}\bigl(\ell_{\nu}(2^n,2^N) -\ell_{\varDelta_M}(2^n,2^N)\bigr)<+\infty.
\end{equation}
\tag{11.5}
$$
Hence by (6.10) of Lemma 3 in the form (6.7), we obtain
$$
\begin{equation*}
\begin{aligned} \, &\limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_{\nu}(2^n,2^N) -J_{i\mathbb{R}}(2^n,2^N;M)\bigr) \\ &\qquad\leqslant \limsup_{N\to \infty}\sup_{0\leqslant n<N} \bigl(\ell_{\nu}(2^n,2^N) -\ell_{\varDelta_M}(2^n,2^N)\bigr) \\ &\qquad\qquad+\limsup_{N\to \infty}\sup_{0\leqslant n<N} |\ell_{\varDelta_M}(2^n,2^N) -J_{i\mathbb{R}}(2^n,2^N;M)| \stackrel{(11.5),(6.10)}{<}+\infty, \end{aligned}
\end{equation*}
\notag
$$
where the extreme parts of these inequalities give (2.15) in assertion II of the main theorem. The implication II $\Rightarrow$ I of the main theorem implies that, for any $b\in [0,s)$, there exists a subharmonic function $U\not\equiv -\infty$ of finite type with Riesz distribution of masses $\frac{1}{2\pi}\Delta U\geqslant \nu$ which satisfies (2.14). This proves the implication II $\Rightarrow$ I, and, therefore, Theorem 10. Theorem 11. Let $\mu$ be a distribution of masses of finite upper density with property [$ \boldsymbol\mu ^{\operatorname{rh}}$]. Suppose that the distribution of points $\mathrm{Z}$ is the same as in Theorem 8. Then the following four assertions I– IV are equivalent. I. For all $0\leqslant b<s\in \mathbb{R}^+$, for an arbitrary subharmonic function $M\not\equiv -\infty$ of finite type with $\frac{1}{2\pi}\Delta M\geqslant\mu$, and, for every subharmonic function $m$ with Riesz distribution of masses $\frac{1}{2\pi}\Delta m= \frac{1}{2\pi}\Delta M\lfloor_{\operatorname{str}_s}$, there exists an entire function $f\not\equiv 0$ with $f(\mathrm{Z})=0$ such that the subharmonic function $\ln |f|+m$ is of finite type and inequalities (2.26) hold. II. If $n\in \mathbb{N}_0$ and $N\in \mathbb{N}$, then
$$
\begin{equation}
\limsup_{N\to \infty}\sup_{0\leqslant n< N} \bigl(\ell_\mathrm{Z}(2^n,2^N) -\ell_{\mu}(2^n,2^N)\bigr)<+\infty.
\end{equation}
\tag{11.6}
$$
III. For all $b\in \mathbb{R}^+$, $d\in (0,2]$, and any function $r\colon \mathbb{C}\to (0,1]$ satisfying (2.17), for every subharmonic function $M\not\equiv -\infty$ of finite type with $\frac{1}{2\pi}\Delta M\geqslant\mu$, there exists an entire function $f\not\equiv 0$ of exponential type with $f(\mathrm{Z})=0$ and $E_b\subset \mathbb{C}$ such that $\ln|f(z)|\leqslant M^{\bullet r}(z)$ for all $z\in \overline{\operatorname{str}}_b$, and also $\ln|f(z)|\leqslant M(z)$ for all $z\in \overline{\operatorname{str}}_b\setminus E_b$, where $E_b$ satisfies (2.20). IV. Assertion V of Theorem 10 holds for the distribution of masses $\nu:=\mathrm{Z}$. We omit the derivation of Theorem 11 from Theorem 10 since it coincides almost verbatim with that of Theorem 8 from the main theorem in § 2.2. The following Corollary 3 can also be deduced from Theorem 11 using a scheme similar to the sequential derivation of Corollaries 1 and 2 from Theorem 8 (see also § 2.2). Corollary 3. If, under the conditions of Theorem 11, a distribution of masses $\mu$ is integer-valued, that is, it is the distribution of points $\mathrm{W}=\mu$, then property [$ \boldsymbol\mu ^{\operatorname{rh}}$] with (10.1) for $r_n\underset{n\in \mathbb{N}_0}{:=}2^n$ is equivalent to property (1.28), and each of the three assertions I– III of Theorem 7 is equivalent to assertion IV of Theorem 11. Remark 12. If, under the conditions of Theorems 10 and 11, property [$ \boldsymbol\mu ^{\operatorname{rh}}$] is replaced by property [$ \boldsymbol\mu ^{\operatorname{lh}}$], then in view of Remark 10, the only change required in the formulation of Theorems 10 and 11 is that to replace in (11.2) the second “right-handed” equality $\varDelta_M\lfloor_{\mathbb{C}_{\mathrm{rh}}}= \mu\lfloor_{\mathbb{C}_{\mathrm{rh}}}$ by the “left-hand side” equality $\varDelta_M\lfloor_{\mathbb{C}_{\mathrm{lh}}}= \mu\lfloor_{\mathbb{C}_{\mathrm{lh}}}$. Accordingly, condition (1.28) for $\mathrm{W}$ in Theorem 7 can be replaced by the $i\mathbb R$-symmetric condition
$$
\begin{equation*}
\limsup_{N\to \infty}\sup_{0\leqslant n<N}\bigl(\ell_\mathrm{W}(2^n,2^N) -\ell_{\mathrm{W}\lfloor_{\mathbb{C}_{\mathrm{lh}}}}(2^n,2^N)\bigr)<+\infty
\end{equation*}
\notag
$$
with $\mathrm{W}\lfloor_{\mathbb{C}_{\mathrm{lh}}}$ $\mathrm{W}$ restricted to $\mathbb{C}_{\mathrm{lh}}$; it is clear that this condition holds for $\mathrm{W}\subset \mathbb{C}_{\overline{\mathrm{lh}}}$. In this case, the only additional necessary change in Theorem 7 is the replacement the right-hand equality $\operatorname{Zero}_g \lfloor_{\mathbb{C}_{\mathrm{rh}}} = \mathrm{W}\lfloor_{\mathbb{C}_{\mathrm{rh}}}$ in assertion III by the left-hand inequality $\operatorname{Zero}_g \lfloor_{\mathbb{C}_{\mathrm{lh}}} =\mathrm{W}\lfloor_{\mathbb{C}_{\mathrm{lh}}}$. 11.2. Concluding remarksWithout touching possible applications of the above results mentioned at the end of the introduction, we point out some possible ways of further natural development of the main results. The Redheffer external density of distribution of point, can be replaced, in the corresponding results, by numerous external densities of Beurling–Malliavin, Kahane, etc. (see [30], [31], [15], [24], [25]), which are known to agree with the Redheffer external density. A detailed study of various relationships between these densities (involving the supplemented and new ones) was given by Krasichkov-Ternovskiĭ [16] with applications to mass distributions. In addition, much more subtle densities and characteristics of distributions of points and masses generated by special classes of test subharmonic functions from the works [34] and [27] can also be used. These classes of test functions are analogues of the classes of test functions of the theory of distributions. These test densities and characteristics would presumably the most natural setting for the theory logarithmic functions of intervals and submeasures for parts of distributions of points or masses near or on the imaginary axis. The approach involving test functions was already found to be useful in the proof of the Beurling–Maliavin theorems (see [17], Ch. III in [26], Theorem 2, and more general versions thereof in [21], when solving other problems in [59], [40], [60], and, more recently, in [61]–[65], etc). The method of test subharmonic functions, which is dual in the potential-theoretic sense to the envelope method from [66], [67], will be capable of taking into account significantly more restrictive (than in Theorems 4, 6, 7 and in the main results from § 2 and § 11) constraints on entire functions of exponential type or subharmonic functions of finite type, with due account of the exact values of their type, or even restrictions on indicators of entire functions. Finally, the problem statements (1.3), (1.6)–(1.8) in the introduction are possible not only for subharmonic functions $M$ of finite type, but also for fairly arbitrary extended numerical functions $M$ defined only on the imaginary axis or on the vertical strip $\overline{\operatorname{str}}_b$ (see (1.5)). These versions of problem statements, which are in the spirit and strictly within the framework of the Beurling–Maliavin theorem on the multiplier, are touched upon in [29], [32], [6], [33]; a thoroughly study can be found in the monographs of Koosis [24], [25], and in some of his subsequent papers. One possible approach in this development is the approximation of extended scalar functions $M$ on the imaginary axis or a strip by subharmonic functions of finite type or by their differences in pointwise, integral, or uniform metrics outside small exceptional sets. This is a different problem of independent interest.
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