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On some potential-theoretic problems related to the asymptotics of Hermite–Padé polynomials N. R. Ikonomova, S. P. Suetinb a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2024, Volume 215, Number 8, DOI: https://doi.org/10.4213/sm10018 
§ 1. Introduction and definitions1.1.Gonchar and Rakhmanov applied in 1981 for the first time the potential-theoretic approach the asymptotic properties of Hermite–Padé polynomials. In [9] (see also [10]) they considered the case of type II Hermite–Padé polynomials for a system of $m$ functions, which also form an Angelesco system. Since $m\geqslant2$ could be an arbitrary natural number, the arising potential-theoretic equilibrium problem is a vector equilibrium problem described in terms of an $ m\times m $ matrix. In 1986 Nikishin [16] introduced a new system of functions which is now called a ‘Nikishin system’ after him. Following the ideas of Gonchar and Rakhmanov [9], Nikishin considered in [16] type I Hermite–Padé polynomials for $m$ functions forming a Nikishin system. The asymptotics of these Hermite–Padé polynomials were described in [16] in terms of an associated vector equilibrium problem, which was stated in terms of an $ m\times m $ matrix. For the further development of this powerful Gonchar–Rakhmanov method, see [1], [15] and the bibliography there. In [18] (see also [5]) the case of two functions forming a Nikishin system was considered. Instead of the vector equilibrium problem for a logarithmic potential, in [18] a scalar equilibrium problem was proposed for a mixed Green-logarithmic potential. The asymptotics of the corresponding type I Hermite–Padé polynomials were described in [18] in terms of the unique solution of this scalar equilibrium problem. In [5] it was proved that for this equilibrium problem there is a conjugate equilibrium problem, which has a unique solution, equal to the balayage of the solution of the original equilibrium problem; for details see [5] and § 1.2 below. It has turned out [25] that the asymptotics of the type II Hermite–Padé polynomials (HP-polynomials) can be described in terms of the unique solution of the conjugate equilibrium problem. It should be noted also that for some special cases the original equilibrium problem was stated and treated for the first time in [11], in connection with the problem of the convergence of the so-called ‘linear’ (or Frobenious-type) and ‘nonlinear’ (or Baker-type) Padé approximations to orthogonal expansions. This is quite natural since there is a well-known connection between type I HP-polynomials and linear Padé approximations to orthogonal expansions and, in particular, expansions in Chebyshev polynomials; see [12], [2], [19] and [26]. In both cases, of the original mixed equilibrium problem and of the conjugate one, the corresponding equilibrium measure $\lambda$ depends on a real nonnegative parameter $\theta$ (see (4) and (5) below), that is, $\lambda=\lambda(\theta)$. Since the rate of convergence of the corresponding constructive rational approximants depends on the Green’s potential of the equilibrium measure $\lambda(\theta)$, it would be natural to consider the property of monotonicity in $\theta$ of Green’s potential to compare these rates with one another. For the initial mixed equilibrium problem this was done in [11] for $\theta=0,1$ and $3$. In this work we consider the case of the conjugate mixed equilibrium problem. We note that the equilibrium problems that come from HP-theory are very closely connected with extremal problems of geometric function theory; see [17], [1] and [18] and cf. [3] and [8]. For the very recent results on the asymptotics of HP-polynomials see [20], [4], [24] and [21]. Note in conclusion that the asymptotic properties of Hermite–Padé polynomials found recently wide applications to the Rayleigh–Schrödinger perturbation theory; see, for example, [6] and [7] and the bibliography there. AcknowledgementThe authors express their gratitude to the reviewers for the careful examination of the present work, and are thankful for their comments, which contributed to the corrections made in the final manuscript. 1.2.For an arbitrary regular (in the sense of the solution of the Dirichlet problem) compact set $K\subset\mathbb{R}$ let us denote by $M_1(K)$ the set of all probability (Borel) measures with support on $K$. For a compact set $S\subset\mathbb{R}$ let $g_S(\zeta,z)$ be the Green’s function for the domain $\widehat{\mathbb{C}}\setminus{S}$ with a logarithmic singularity at the point $\zeta=z$. Let $V^\mu(z)$ be the logarithmic potential of the measure $\mu\in M_1(K)$ and $G^\mu_S(z)$ be the Green’s potential of $\mu$ corresponding to the Green’s function $g_S(\zeta,z)$, $S\cap K=\varnothing$:
$$
\begin{equation}
V^\mu(z):=\int_K\log\frac1{|z-t|}\,d\mu(t) \quad\text{and}\quad G^\mu_S(z):=\int_K g_S(t,z)\,d\mu(t).
\end{equation}
\tag{1}
$$
Recall that for a positive Borel measure $\varkappa$ such that $\operatorname{supp}{\varkappa}\Subset\mathbb{R}$,
$$
\begin{equation*}
\widehat{\varkappa}(z):=\int\frac{d\varkappa(x)}{z-x}, \qquad z\in\widehat{\mathbb C}\setminus\operatorname{supp}{\varkappa},
\end{equation*}
\notag
$$
is a Markov-type function. It is well known (see [14]) that there exists a unique measure $\tau_K\in M_1(K)$ with the following property:
$$
\begin{equation}
V^{\tau_K}(x)\equiv\gamma_K=\mathrm{const}, \qquad x\in K.
\end{equation}
\tag{2}
$$
The measure $\tau_K$ is called the Robin measure (or the equilibrium measure) for the compact set $K$, and $\gamma_K$ is the Robin constant of $K$. Notice that $\operatorname{supp}{\tau_K}=K$. Let
$$
\begin{equation*}
g_K(z,\infty)=\log|z|+\gamma_K+o(1),\qquad z\to\infty,
\end{equation*}
\notag
$$
be the Green’s function for the domain $D=D(K):=\widehat{\mathbb{C}}\setminus{K}$ with a logarithmic singularity at the point at infinity $z=\infty$. Then Let $E\subset\mathbb{R}$ and $F\subset\mathbb{R}$ be two compact sets consisting of a finite number of disjoint closed intervals, $E=\bigsqcup_{j=1}^p E_j$ and $F=\bigsqcup_{k=1}^qF_k$, and such that $\operatorname{conv}(E)\cap \operatorname{conv}(F)=\varnothing$, where $\operatorname{conv}(\,{\cdot}\,)$ denotes the convex hull of the corresponding real set. For definiteness we suppose that $F$ is positioned to the right of $E$ (see Figure 1). We note that the ordered pair of compact sets $(E,F)$ forms a so-called Nuttall condenser (see [18], where this notion was first introduced). In the asymptotic theory of the Hermite–Padé polynomials for a pair of functions that form a Nikishin system, the Nuttall condenser plays the same role as the Stahl compact set plays in the theory of Padé polynomials. From the equilibrium equations (4) and (5) it follows that we have indeed an ordered pair of compact sets. For a nonnegative real $\theta$ consider the mixed Green-logarithmic potential ${\theta V^\mu(z) +G^\mu_F(z)}$, where $\mu\in M_1(E)$. It is well known (see [11], [18] and [5]) that there exists a unique measure $\lambda_E=\lambda_E(\theta)$ with support on $E$, $\lambda_E(\theta)\in M_1(E)$, that has the following property:
$$
\begin{equation}
\theta V^{\lambda_E}(x)+G^{\lambda_E}_F(x) \equiv c_E(\theta)=\mathrm{const}, \qquad x\in E.
\end{equation}
\tag{4}
$$
Notice that $\operatorname{supp}{\lambda_E}=E$. Let us introduce another mixed Green-logarithmic potential, namely, ${\theta V^\nu(z) +G^\nu_E(z)}$, $\nu\in M_1(F)$, and consider for it the following equilibrium problem with the potential of an external field given by $g_E(z,\infty)$:
$$
\begin{equation}
\theta V^{\lambda_F}(y)+G^{\lambda_F}_E(y) +\theta g_E(y,\infty)\equiv c_F(\theta)=\mathrm{const}, \qquad y\in F, \qquad \lambda_F\in M_1(F).
\end{equation}
\tag{5}
$$
It is known (see [5]) that there exists a unique solution of (5), which is a measure $\lambda_F=\lambda_F(\theta)$ with support on $F$, $\lambda_F(\theta)\in M_1(F)$. Notice that $\operatorname{supp}{\lambda_F}=F$. The following connection1 (see [5]) exists between $\lambda_E(\theta)\in M_1(E)$ and ${\lambda_F(\theta)\in M_1(F)}$: where ${\mathfrak b}_F(\,{\cdot}\,)$ is the balayage of a given measure from the domain $G:=\widehat{\mathbb{C}}\setminus{F}$ onto its boundary $\partial G=F$.Also, the following connection exists between the potentials:
$$
\begin{equation}
\theta V^{\lambda_F(\theta)}(z)+G^{\lambda_F(\theta)}_E(z) +\theta g_E(z,\infty)\equiv -(1+\theta)G^{\lambda_E(\theta)}_F(z)+c_F(\theta), \qquad z\in\widehat{\mathbb C}
\end{equation}
\tag{7}
$$
(see [5], formula (24)). In view of relation (6) and identity (7), the potential-theoretic equilibrium problem (5) can be considered to be conjugate to the equilibrium problem (4).
§ 2. Statement of the problem2.1.It was proved in [11] (see also [12]) that the Green’s potentials of the equilibrium measures $\lambda_F(1)$ and $\lambda_F(3)$ determine the rates of convergence of the nonlinear ($F_n$) and linear ($\Phi_n$) Padé approximants, respectively, to the orthogonal expansion of a Markov-type function $f(z)=\widehat{\varkappa}(z)$ corresponding to $\operatorname{supp}{\varkappa}=F=[c,d]$, $b<c$. More precisely, let $N=2n+1=3m+1$ and let $c_0,c_1,\dots, c_{N-1}$ be the $N$ Fourier coefficients of the orthogonal expansion of the Markov-type function $f(z)$ with respect to a measure $\sigma$ on $E=[a,b]$ such that $\sigma'=d\sigma/dx>0$ almost everywhere on $[a,b]$. Then from these $N$ Fourier coefficients we can construct the nonlinear Padé–Fourier approximant $F_n$ of degree $(n,n)$ and the linear Padé–Fourier approximant $\Phi_m$ of degree $(m,m)$, where $3m=2n$. It was proved in [11] that the rates of convergence of the constructive (with respect to the given $c_0,c_1,\dots,c_{N-1}$; see [13], § 2) rational approximants $R_N=F_n$ and $R_N=\Phi_m$ are determined by the following relations, which take place locally uniformly2 in the domain $G$:
$$
\begin{equation}
\frac1N\log|f(z)-R_N(z)| = \begin{cases} \delta_1(z):=-\dfrac23\,G^{\lambda_E(3)}_F(z) &\text{for }R_N=\Phi_m, \\ \delta_2(z):=-G^{\lambda_E(1)}_F(z) &\text{for }R_N=F_n. \end{cases}
\end{equation}
\tag{8}
$$
Since $G^{\lambda_E(1)}_F(z) >\frac23G^{\lambda_E(3)}_F(z)$ for $z\in G$ (see [11], formula (9)), we have ${\delta_2(z)\,{<}\,\delta_1(z)}$ and thus, the nonlinear Padé–Fourier approximants $F_n$ are more efficient than the linear Padé–Fourier approximants $\Phi_m$ as $N\to\infty$. Notice (see [11]) that the equilibrium measure $\lambda_E(0)$ and the corresponding function $G^{\lambda_E(0)}_F$ are associated with the best (Chebyshev) rational approximants of degree $(n,n)$ to the Markov type function $f(z)=\widehat{\varkappa}(z)$ on $E=[a,b]$. However, these approximants are not constructive in the sense of the paper [13] by Henrici (see [13], § 2). 2.2.Now let $f(z)\in\mathscr{H}(\infty)$ be given by the following explicit representation:
$$
\begin{equation}
f(z):=\biggl[\biggl(A-\frac1{\varphi(z)}\biggr) \biggl(B-\frac1{\varphi(z)}\biggr)\biggr]^{-1/2}, \qquad z\notin[-1,1],
\end{equation}
\tag{9}
$$
where $1<A<B$ and the branch of the root function $(\,{\cdot}\,)^{1/2}$ is chosen so that $\varphi(z)=z+(z^2-1)^{1/2}\sim 2z$ and, as a result, $f(z)\sim1/\sqrt{AB}$ as $z\to\infty$. The function $f$ is an algebraic function of fourth degree with four branch points $\{\pm1,a,b\}$, where $a=(A+1/A)/2$ and $b=(B+1/B)/2$, $1<a<b$. The interval $E=[-1,1]$ is the Stahl compact set $S(f)$ for $f$ given by (9) and $D:=\widehat{\mathbb{C}}\setminus{E}$ is the corresponding Stahl domain. We denote the class of such analytic functions by $\mathscr Z([-1,1])$. In [23] it was proved that $f\in\mathscr Z([-1,1])$ is a Markov-type function, and the pair $f$, $f^2$ and the triple $f$, $f^2$, $f^3$ form Nikishin systems [16], [17]:
$$
\begin{equation*}
\begin{gathered} \, f(z)=\frac1{\sqrt{AB}}+\widehat{\sigma}(z), \qquad \operatorname{supp}{\sigma}=E, \\ f^2(z)=\frac1{AB}+\frac1{\sqrt{AB}}\,\widehat{\sigma}(z) +\widehat{s}_1(z), \qquad \operatorname{supp}{s_1}=E, \\ f^3(z)=\frac1{\sqrt{(AB)^3}}+\frac1{AB}\,\widehat{\sigma}(z) +\frac1{\sqrt{AB}}\,\widehat{s}_1(z)+\widehat{s}_2(z), \qquad \operatorname{supp}{s_2}=E, \end{gathered}
\end{equation*}
\notag
$$
where $\operatorname{supp}\sigma=\operatorname{supp}{s}_1=\operatorname{supp}{s}_2=E$, $s_1:=\langle \sigma,\sigma_2\rangle$, that is, $ds_1(z)=\widehat{\sigma}_2(z)\,d\sigma(z)$, $\operatorname{supp}{\sigma_2}=[a,b]$ and $s_2:=\langle \sigma,\langle \sigma_2,\sigma\rangle\rangle$. The measures $\sigma$ and $\sigma_2$ admit explicit representations; see [23], formulae (16) and (17). Let $f\in\mathscr Z([-1,1])$, $f\in\mathscr{H}(\infty)$ and $N=2n+1=3m+1=4\ell+1$. For the pair of functions $f$, $f^2$ let us define the type II HP-polynomials $P^{(2)}_{2m,0}\not\equiv0$, $P^{(2)}_{2m,1}$ and $P^{(2)}_{2m,2}$ of degree $\leqslant 2m$ by the relations
$$
\begin{equation}
\begin{gathered} \, \bigl(P^{(2)}_{2m,0}f-P^{(2)}_{2m,1}\bigr)(z)=O(z^{-m-1}), \qquad z\to\infty, \\ \bigl(P^{(2)}_{2m,0}f^2-P^{(2)}_{2m,2}\bigr)(z) =O(z^{-m-1}), \qquad z\to\infty. \end{gathered}
\end{equation}
\tag{10}
$$
Analogously, for the triple of functions $f$, $f^2$, $f^3$ let us define the type II HP-polynomials $P^{(3)}_{3\ell,0}\not\equiv0$, $P^{(3)}_{3\ell,1}$, $P^{(3)}_{3\ell,2}$ and $P^{(3)}_{3\ell,3}$, $\operatorname{deg}P^{(3)}_{3\ell,j}\leqslant 3\ell$, by the relations:
$$
\begin{equation}
\begin{gathered} \, \bigl(P^{(3)}_{3\ell,0}f-P^{(3)}_{3\ell,1}\bigr)(z) =O(z^{-\ell-1}), \qquad z\to\infty, \\ \bigl(P^{(3)}_{3\ell,0}f^2-P^{(3)}_{3\ell,2}\bigr)(z) =O(z^{-\ell-1}), \qquad z\to\infty, \\ \bigl(P^{(3)}_{3\ell,0}f^3-P^{(3)}_{3\ell,3}\bigr)(z)=O(z^{-\ell-1}), \qquad z\to\infty. \end{gathered}
\end{equation}
\tag{11}
$$
For an arbitrary polynomial $Q\not\equiv0$ set where each zero $\eta$ of $Q$ is counted in accordance with its multiplicity.From the results of [1] it follows that, as $N\to\infty$,
$$
\begin{equation*}
\frac1{2m}\chi(P^{(2)}_{2m,j})\xrightarrow{*} \lambda_E(3)\quad \text{and}\quad \frac1{3\ell}\chi(P^{(3)}_{3\ell,k})\xrightarrow{*} \lambda_E(1),
\end{equation*}
\notag
$$
where ‘$\xrightarrow{*}$’ denotes convergence in the space $M_1(E)$ in the weak-$*$ topology, ${j\!=\!0,1,2}$ and $k=0,1,2,3$. 2.3.In [25] (see also [6]) the following result was announced. Theorem 1 ([25]). Let $f\in\mathscr Z(E)$, $E=[-1,1]$. Then, as $N\to\infty$, the following relations are valid locally uniformly in the domain $D=\widehat{\mathbb{C}}\setminus{E}$: and From Stahl’s theory [22] it follows that locally uniformly in $D$
$$
\begin{equation}
\lim_{N\to\infty}\frac1N\log \bigl|f(z)-[n/n]_f(z)\bigr|=-g_E(z,\infty)=:\delta_1(z)<1,
\end{equation}
\tag{14}
$$
where $[n/n]_f$ is the $n$th diagonal Padé approximant to the function $f(z)$. Thus $\delta_2(z)<\delta_1(z)<1$. Therefore, if we prove that
$$
\begin{equation}
G^{\lambda_F(1)}_E(z)>\frac23 G^{\lambda_F(3)}_E(z), \qquad z\in D,
\end{equation}
\tag{15}
$$
then from (12)–(15) we finally obtain
$$
\begin{equation}
\delta_3(z)<\delta_2(z)<\delta_1(z)<1, \qquad z\in D.
\end{equation}
\tag{16}
$$
The main result of this paper is the following statement, which defines the monotonicity property of the potential $G_E^{\lambda_F(\theta)}(z)$ with equilibrium measure ${\lambda_F(\mkern-1mu\theta\mkern-1mu)\mkern-1mu\!\in\! M_1(\mkern-1mu F\mkern-1mu)}$ for $\theta\in[1,3]$. Relation (15) follows directly from (17) for $\theta_1=1$ and $\theta_2=3$. § 3. Proof of Theorem 23.1.Let $\mathfrak{R}_3$ be the three-sheeted Riemann surface constructed in the following way (see Figure 1). The zeroth sheet $\mathfrak{R}^{(0)}_3$ of the Riemann surface $\mathfrak{R}_3$ is a copy of the extended complex plane (Riemann sphere) $\widehat{\mathbb{C}}$, cut along the compact set $F=\bigsqcup_{k=1}^qF_k$. Thus we consider this new compact set as two-sided and denote it by $F^{(0)}_{+}\cup F^{(0)}_{-}$. The first sheet $\mathfrak{R}^{(1)}_3$ is constructed in a similar way: we consider a copy of the extended complex plane $\widehat{\mathbb{C}}$ cut along the two compact sets $E=\bigsqcup_{j=1}^pE_j$ and $F=\bigsqcup_{k=1}^qF_k$. Both these new compact sets are considered as two-sided and denoted by $E^{(1)}_{+}\cup E^{(1)}_{-}$ and $F^{(1)}_{+}\cup F^{(1)}_{-}$, respectively. Finally, to obtain the second sheet $\mathfrak{R}^{(2)}_3 $ we take a copy of the extended complex plane $\widehat{\mathbb{C}}$ cut along the compact set $E=\bigsqcup_{j=1}^pE_j$, consider this new compact set as two-sided and denote it by $E^{(2)}_{+}\cup E^{(2)}_{-}$. Finally, we construct the whole Riemann surface $\mathfrak{R}_3$ by gluing crosswise3 the zeroth sheet $\mathfrak{R}^{(0)}_3$ with the first sheet $\mathfrak{R}^{(1)}_3$ along $F^{(0)}_{+}\cup F^{(0)}_{-}$ and $F^{(1)}_{+}\cup F^{(1)}_{-}$ and gluing crosswise the second sheet $\mathfrak{R}^{(2)}_3$ with the first sheet $\mathfrak{R}^{(1)}_3$ along $E^{(2)}_{+}\cup E^{(2)}_{-}$ and $E^{(1)}_{+}\cup E^{(1)}_{-}$. We denote the point on $\mathfrak{R}^{(0)}_3$ corresponding to a point $z$ in $G=\widehat{\mathbb{C}}\setminus{F}$ by $z^{(0)}$, the point on $\mathfrak{R}^{(1)}_3$ corresponds to $z$ in $\widehat{\mathbb{C}}\setminus({E\cup F})$ by $z^{(1)}$ and the point on $\mathfrak{R}^{(2)}_3$ corresponding to $z$ in $D=\widehat{\mathbb{C}}\setminus{E}$ by $z^{(2)}$. Set $F^{(2)}:=\{z^{(2)}\colon z\in F\}$ and $E^{(0)}:=\{z^{(0)}\colon z\in E\}$. The boundary $\Gamma^{(0,1)}$ between the zeroth and first sheets consists of $q$ closed disjoint curves on the Riemann surface $\mathfrak{R}_3$: $\Gamma^{(0,1)}=\bigsqcup_{k=1}^q \Gamma^{(0,1)}_k$. Similarly, the boundary $\Gamma^{(1,2)}$ between the first and second sheets consists of $p$ closed disjoint curves on $\mathfrak{R}_3$: $\Gamma^{(1,2)}=\bigsqcup_{j=1}^p \Gamma^{(1,2)}_j$. For a point on $\mathfrak{R}_3$ we use the notation $\mathbf{z}$. In particular, $\mathbf{z}=z^{(0)}\in\mathfrak{R}^{(0)}_3$, $\mathbf{z}=z^{(1)}\in\mathfrak{R}^{(1)}_3$ and $\mathbf{z}=z^{(2)}\in\mathfrak{R}^{(2)}_3$. Thus, there is a natural projection $\pi$ of $\mathfrak{R}_3$ onto $\widehat{\mathbb{C}}$, $\pi\colon\mathfrak{R}_3\to \widehat{\mathbb{C}}$, given by the equality $\pi(z^{(j)})=z$, $j=0,1,2$, for $z^{(j)}\in\mathfrak{R}^{(j)}_3$ and $z\notin(E\cup F)$. For the disjoint curves $\Gamma^{(0,1)}$ and $\Gamma^{(1,2)}$ on $\mathfrak{R}_3$ we have $\pi(\Gamma^{(0,1)})=F$ and $\pi(\Gamma^{(1,2)})=E$, respectively. Therefore, $\pi(\mathbf{z})\in F$ for $\mathbf{z}\in\Gamma^{(0,1)}$ and $\pi(\mathbf{z})\in E$ for $\mathbf{z}\in\Gamma^{(1,2)}$. For points $\mathbf z\in\mathfrak R_3\setminus\bigl(\Gamma^{(0,1)}\cup\Gamma^{(1,2)}\bigr)$ we define a $*$-operation $\mathbf{z}\mapsto\mathbf{z}_{*}$ by the following rule:
$$
\begin{equation}
\mathbf z_{*}:=(\overline{z})^{(j)} \quad\text{for } \mathbf z=z^{(j)}, \qquad j=0,1,2,
\end{equation}
\tag{18}
$$
where $\overline{z}$ is the complex conjugate of $z\in\mathbb{C}$. Figure 1 shows the three-sheeted Riemann surface $\mathfrak{R}_3$ constructed, along with the ‘physical’ complex plane $\widehat{\mathbb{C}}$ which contains the compact sets $E$ and $F$. We denote by $E^{(0)}\subset\mathfrak{R}^{(0)}_3$ the compact set corresponding to $E=\bigsqcup_{j=1}^pE_j\subset\widehat{\mathbb{C}}$ and by $F^{(2)} \subset \mathfrak{R}^{(2)}_3$ the compact set corresponding to $F\subset\widehat{\mathbb{C}}$. Note that all three sheets of Riemann surface $\mathfrak{R}_3$ are open subsets of $\mathfrak{R}_3$ and $\mathfrak{R}_3=\mathfrak{R}_3^{(0)} \sqcup\Gamma^{(0,1)}\sqcup\mathfrak{R}_3^{(1)} \sqcup\Gamma^{(1,2)}\sqcup\mathfrak{R}_3^{(2)}$. 3.2.We define a function $v_\theta(\mathbf{z})$ in the following way:
$$
\begin{equation}
v_\theta(\mathbf z):= \begin{cases} (\theta+1)G^{\lambda_F(\theta)}_E(z), &\mathbf z=z^{(0)}, \\ -2\theta V^{\lambda_F(\theta)}(z)+(\theta-1)G^{\lambda_F(\theta)}_E(z)-2\theta g_E(z,\infty)+2c_F(\theta), &\mathbf z=z^{(1)}, \\ -2\theta V^{\lambda_F(\theta)}(z)-(\theta-1)G^{\lambda_F(\theta)}_E(z)+2\theta g_E(z,\infty)+2c_F(\theta), &\mathbf z=z^{(2)}. \end{cases}
\end{equation}
\tag{19}
$$
Clearly, $v_\theta(\mathbf{z})\equiv0$ for $\mathbf{z}=z^{(0)}\in E^{(0)}$, $v_\theta(\mathbf{z})=4\theta\log|z|+O(1)$ as $\mathbf{z}\to\infty^{(2)}$, and $v_\theta(\mathbf{z})$ is a harmonic function on $\mathfrak{R}^{(0)}_3\setminus E^{(0)}$, $\mathfrak{R}^{(1)}$ and $\mathfrak{R}^{(2)}_3\setminus(F^{(2)}\cup\infty^{(2)})$. Thus, $v_\theta(\mathbf{z})$ is a piecewise harmonic function except at the point $\mathbf{z}=\infty^{(2)}$. Lemma 1. The piecewise harmonic function $v_\theta(\mathbf{z})$ can be extended across the curves $\Gamma^{(0,1)}$ and $\Gamma^{(1,2)}$ to produce a harmonic function on $\mathfrak{R}_3\setminus(E^{(0)}\cup F^{(2)}\cup\infty^{(2)})$. Proof. Let $ U^{(0,1)}\subset\mathfrak{R}_3$ be a neighbourhood of $\Gamma^{(0,1)}$ such that $ U^{(0,1)}\cap E^{(0)}=\varnothing$ and $U^{(0,1)}\cap \Gamma^{(1,2)}=\varnothing$; see Figure 1. From (5) it follows that is a harmonic function in $ U^{(0)}:= U^{(0,1)}\cap \mathfrak{R}^{(0)}_3$ with the following property: it is continuous on $U^{(0)}\cup \Gamma^{(0,1)}$ and $u_1(z^{(0)})\equiv0$ for $z^{(0)}\in\Gamma^{(0,1)}$. Therefore, we can continue the function $u_1(\mathbf{z})$, $\mathbf{z}\in U^{(0)}$, across the curves $\Gamma^{(0,1)}$ to the open set $U^{(1)}:=U^{(0,1)}\cap \mathfrak{R}^{(1)}_3$ with opposite values at $*$-symmetric points: The extended function $u_1(\mathbf{z})$ is a harmonic function in $U^{(0,1)}\supset\Gamma^{(0,1)}$.
Let $\widetilde{\lambda}={\mathfrak b}_E(\lambda_F(\theta))\in M_1(E)$ be the balayage of the measure $\lambda_F(\theta)\in M_1(F)$ from the domain $D$ onto its boundary $\partial{D}=E$. Then $V^{\widetilde{\lambda}}(z)\equiv V^{\lambda_F(\theta)}(z)-G_E^{\lambda_F(\theta)}(z)+ \mathrm{const}$, $z\in\widehat{\mathbb C}$. Since $\widetilde{\lambda}\in M_1(E)$, the potential $V^{\widetilde{\lambda}}(z)$ is a harmonic function in $\pi(U^{(0,1)})$. The function $V^{\widetilde{\lambda}}$ can be lifted to the set $U^{(0,1)}\subset\mathfrak{R}_3$, as well as to the sheets $\mathfrak{R}^{(0)}_3$, $\mathfrak{R}^{(1)}_3$ and $\mathfrak{R}^{(2)}_3$ in the following way:
$$
\begin{equation}
V^{\widetilde{\lambda}}(\mathbf z):=V^{\widetilde{\lambda}}(\pi(\mathbf z))= V^{\widetilde{\lambda}}(z).
\end{equation}
\tag{20}
$$
The same is true for the Green’s function $g_E(z,\infty)$:
$$
\begin{equation}
g_E(\mathbf z,\infty):=g_E(\pi(\mathbf z),\infty)=g_E(z,\infty).
\end{equation}
\tag{21}
$$
From (20) and (21) it follows that we can define the new function $u_2(\mathbf z):=u_1(\mathbf z)-\theta V^{\widetilde{\lambda}}(\mathbf z)-\theta g_E(\mathbf z,\infty)+\mathrm{const}$, which is harmonic for $\mathbf{z}\in U^{(0,1)}$, and we obtain the following representation in $U^{(0,1)}$:
$$
\begin{equation}
u_2(\mathbf z)= \begin{cases} (\theta+1)G^{\lambda_F(\theta)}_E(z)-c_F(\theta), & \mathbf z=z^{(0)}, \\ -2\theta V^{\lambda_F(\theta)}(z)+(\theta-1)G^{\lambda_F(\theta)}_E(z)-2\theta g_E(z,\infty)+2c_F(\theta), & \mathbf z=z^{(1)}. \end{cases}
\end{equation}
\tag{22}
$$
This function $u_2(\mathbf{z})$ is harmonic in $U^{(0,1)}$.
Let $U^{(1,2)}\subset\mathfrak{R}_3$ be a neighbourhood of $\Gamma^{(1,2)}$ such that $U^{(1,2)}\cap \Gamma^{(0,1)}=\varnothing$ and $U^{(1,2)}\cap F^{(2)}=\varnothing$. Since $\lambda_F(\theta)\in M_1(F)$, the function $V^{\lambda_F(\theta)}(z)$ is harmonic in $\pi(U^{(1,2)})$. Also, $G^{\lambda_F(\theta)}_E(z)\equiv0$ and $g_E(z,\infty)\equiv0$ for $z\in E$, so that the function
$$
\begin{equation}
\begin{aligned} \, u_3(\mathbf z) &:=-2\theta V^{\lambda_F(\theta)}(z)+(\theta-1)G^{\lambda_F(\theta)}_E(z) \nonumber \\ &\qquad-2\theta g_E(z,\infty) +2c_F(\theta), \qquad \mathbf z\in U^{(1,2)}\cap\mathfrak R^{(1)}_3, \end{aligned}
\end{equation}
\tag{23}
$$
can be continued across the curve $\Gamma^{(1,2)}$ to the open set $U^{(1,2)}\cap\mathfrak{R}^{(2)}_3$ as a harmonic function in the following way:
$$
\begin{equation}
\begin{aligned} \, u_3(\mathbf z) &=-2\theta V^{\lambda_F(\theta)}(z)-(\theta-1)G^{\lambda_F(\theta)}_E(z) \nonumber \\ &\qquad+2\theta g_E(z,\infty) +2c_F(\theta), \qquad \mathbf z\in U^{(1,2)}\cap\mathfrak R^{(2)}_3. \end{aligned}
\end{equation}
\tag{24}
$$
The functions $u_2(\mathbf{z})$ and $u_3(\mathbf{z})$ (see (22)–(24)) in combination give us the function $v_\theta(\mathbf{z})$ (see (19)). Thus, Lemma 1 is proved. 3.3.Since we have the identity
$$
\begin{equation*}
\begin{aligned} \, \theta V^{\lambda_F(\theta)}(z)-c_F(\theta) &\equiv \bigl(\theta V^{\lambda_F(\theta)}(z)+G^{\lambda_F(\theta)}_E(z)+\theta g_E(z,\infty)-c_F(\theta)\bigr) \\ &\qquad-G^{\lambda_F(\theta)}_E(z)-\theta g_E(z,\infty), \end{aligned}
\end{equation*}
\notag
$$
from (5) and the definition (19) we obtain
$$
\begin{equation}
\begin{aligned} \, v_\theta(z^{(2)}) &=-2\bigl(\theta V^{\lambda_F(\theta)}(z)+G^{\lambda_F(\theta)}_E(z)+\theta g_E(z,\infty)-c_F(\theta)\bigr) \notag \\ &\qquad+2G^{\lambda_F(\theta)}_E(z)+4\theta g_E(z,\infty) -(\theta-1)G^{\lambda_F(\theta)}_E(z) \notag \\ &=(3-\theta) G^{\lambda_F(\theta)}_E(z)+4\theta g_E(z,\infty), \qquad z\in F. \end{aligned}
\end{equation}
\tag{25}
$$
Set
$$
\begin{equation}
v(\mathbf z)=\frac1{\theta_1}\, v_{\theta_1}(\mathbf z)-\frac1{\theta_2}\, v_{\theta_2}(\mathbf z).
\end{equation}
\tag{26}
$$
The function $v(\mathbf{z})$ is a continuous function on the Riemann surface $\mathfrak{R}_3$ and is a harmonic function in the domain $\mathfrak{R}_3\setminus(E^{(0)}\cup F^{(2)})$. From (25) and (26) it follows that
$$
\begin{equation}
\begin{aligned} \, v(z^{(0)}) &=\biggl(1+\frac1{\theta_1}\biggr)G^{\lambda_F(\theta_1)}_E(z) -\biggl(1+\frac1{\theta_2}\biggr)G^{\lambda_F(\theta_2)}_E(z), \qquad z\in \widehat{\mathbb C}\setminus {F}, \\ v(z^{(2)}) &= \biggl(\frac3{\theta_1}-1\biggr)G^{\lambda_F(\theta_1)}_E(z) -\biggl(\frac3{\theta_2}-1\biggr)G^{\lambda_F(\theta_2)}_E(z), \qquad z\in F. \end{aligned}
\end{equation}
\tag{27}
$$
Note that for $\theta_1=1$ and $\theta_2=3$, from (27) we obtain
$$
\begin{equation*}
v(z^{(2)})=2 G^{\lambda_F(1)}_E(z)>0, \qquad z^{(2)}\in F^{(2)}.
\end{equation*}
\notag
$$
From here and (27), by the minimum principle for harmonic functions it follows that $v(\mathbf{z})>0$ in the domain $\mathfrak D:=\mathfrak{R}_3\setminus (E^{(0)}\cup F^{(2)})$. Therefore,
$$
\begin{equation*}
2G^{\lambda_F(1)}_E(z)-\frac43\, G^{\lambda_F(3)}_E(z)>0, \qquad z\in \widehat{\mathbb C}\setminus E.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
G^{\lambda_F(1)}_E(z)>\frac23\, G^{\lambda_F(3)}_E(z), \qquad z\in \widehat{\mathbb C}\setminus E,
\end{equation*}
\notag
$$
and Theorem 2 follows for $\theta_1=1$ and $\theta_2=3$. Clearly, the theorem is also true for ${1<\theta_1<\theta_2=3}$. Now we prove the theorem in the general case of $1\leqslant\theta_1<\theta_2<3$. If $v(z^{(2)})\geqslant0$ for $z^{(2)}\in F^{(2)}$, then $v(\mathbf{z})>0$ in the domain $\mathfrak D$, since $v(z^{(0)})\equiv0$ on $E^{(0)}$, and it is easy to check that $v(\mathbf{z})\not\equiv0$. Theorem 2 follows from the inequality $v(\mathbf{z})>0$ in the domain $\mathfrak D$. Now assume that $m:=\min_{z\in F}v(z^{(2)})<0$. Then there exists a point $y_0\in F$ such that $m=v(y^{(2)}_0)<0$. Therefore,
$$
\begin{equation}
\biggl(\frac3{\theta_1}-1\biggr)G^{\lambda_F(\theta_1)}_E(y_0)< \biggl(\frac3{\theta_2}-1\biggr)G^{\lambda_F(\theta_2)}_E(y_0).
\end{equation}
\tag{28}
$$
From the minimum principle for harmonic functions it follows that $v(\mathbf{z})>m$ for ${\mathbf{z}\in\mathfrak D}$. Hence $v(y_0^{(0)})>m$, and from the definition (26) of the function $v(\mathbf{z})$ and (27) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl(\frac1{\theta_1}+1\biggr)G^{\lambda_F(\theta_1)}_E(y_0)- \biggl(\frac1{\theta_2}+1\biggr)G^{\lambda_F(\theta_2)}_E(y_0) \\ &\qquad >m= \biggl(\frac3{\theta_1}-1\biggr)G^{\lambda_F(\theta_1)}_E(y_0) -\biggl(\frac3{\theta_2}-1\biggr)G^{\lambda_F(\theta_2)}_E(y_0). \end{aligned}
\end{equation}
\tag{29}
$$
The relation (29) implies the following inequality:
$$
\begin{equation}
\biggl(2-\frac2{\theta_1}\biggr)G^{\lambda_F(\theta_1)}_E(y_0)> \biggl(2-\frac2{\theta_2}\biggr)G^{\lambda_F(\theta_2)}_E(y_0), \qquad y_0\in F.
\end{equation}
\tag{30}
$$
In the case when $\theta_1=1$ relation (30) implies a contradiction since $G^{\lambda_F(\theta_2)}_E(y_0)>0$. Thus, we can now assume that $1<\theta_1<\theta_2<3$. Dividing (28) by (30) we obtain
$$
\begin{equation}
\biggl(\frac3{\theta_1}-1\biggr)\biggl(2-\frac2{\theta_1}\biggr)^{-1}< \biggl(\frac3{\theta_2}-1\biggr)\biggl(2-\frac2{\theta_2}\biggr)^{-1}.
\end{equation}
\tag{31}
$$
Inequality (31) implies that
$$
\begin{equation}
\frac{3-\theta_1}{\theta_1-1}<\frac{3-\theta_2}{\theta_2-1},
\end{equation}
\tag{32}
$$
where $1<\theta_1<\theta_2<3$. Let the function $g(t)$ be given by the representation It is easy to check that Therefore, $g(t)$ is a decreasing function on the interval $(1,3)$. This contradicts (32). Relation (32) holds under the assumption that $\min_{z\in F}v(z^{(2)})<0$. Therefore, it follows from this contradiction that $v(z^{(2)})\geqslant0$ on $F^{(2)}$. Theorem 2 is proved.
Citation in format AMSBIB
\Bibitem{IkoSue24} |
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