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This article is cited in 4 scientific papers (total in 4 papers) A direct proof of Stahl's theorem for a generic class of algebraic functions S. P. Suetin Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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     § 1. Introduction1.1Stahl’s seminal theory on the convergence of the diagonal Padé approximants for multivalued analytic functions (see [17], [18], [1] and the bibliography there) consists of two parts, the geometric part and the analytic one. In the first, geometric, part it is proved that, given a multivalued function, there exists a unique admissible compact set $S$ possessing the $S$-property or, briefly, an $S$-compact set. In Stahl’s general theory one assumes that the set of singularities of the multivalued analytic function has zero (logarithmic) capacity. Here we restrict our investigations to the case when there is a finite number of singular points. Note that a short proof of the existence of an $S$-compact set in this case was proposed by Rakhmanov (1994) in the unpublished paper [11] (also see [8]). Rakhmanov’s proof is based on the connections between the capacity and the potential of the equilibrium measure of a compact set. In the second, analytic, part of Stahl’s theory it is proved that the Padé polynomials have a limiting distribution of zeros and the diagonal Padé approximants converge in capacity. Stahl’s original proof of the existence of a limiting distribution of the Padé polynomials is based on the properties of $S$-compact sets and goes by contradiction. In this paper we give a short and direct proof of the second part of Stahl’s theory for a generic class of algebraic functions satisfying certain conditions, under the assumption that the first geometric component of the theory is valid. We do not use the relations of orthogonality for Padé polynomials. The proof is based only on the maximum principle for subharmonic functions (cf. [10] and [15]). Let introduce the class $\mathscr F$ of admissible multivalued functions under consideration in our paper. For a multivalued function $f$ we agree that $f$ belongs to $\mathscr F$ if the following conditions are satisfied: (I) $f$ is an algebraic function all of whose branch points have the second order (that is, we assume that all branch points are of square root type); (II) there exists a germ $f_\infty\in\mathscr H(\infty)$ of $f$ such that the $S$-compact set for it consists of a finite number of disjoint closed analytic arcs each of which contains exactly two branch points of $f$, the endpoints of the arc. If conditions (I) and (II) are satisfied, then we write $(f,f_\infty)\in\mathscr F$. Some discussions relating to assumptions (I) and (II) are presented in § 3 (also see [12], [9], [10] and [6]). Note the following: condition (II) means that $S$ contains no Chebotarev points (cf. [16] and [2]). 1.2Let $f_\infty\in\mathscr H(\infty)$, and let $S=S(f_\infty)$ be Stahl’s $S$-compact set corresponding to the germ $f_\infty$. Let $\lambda_S$ be the probability equilibrium measure with support on $S$ and $V^{\lambda_S}(z)$ be the corresponding logarithmic potential of $\lambda_S$:
$$
\begin{equation}
V^{\lambda_S}(z)=\gamma_S-g_S(z,\infty), \qquad z\in D:=\widehat{\mathbb C}\setminus S,
\end{equation}
\tag{1}
$$
where $g_S(z,\infty)$ is the Green’s function for the Stahl domain $D=\widehat{\mathbb C}\setminus{S}$ with logarithmic singularity at the point at infinity $z=\infty$, and $\gamma_S$ is the Robin constant for $D$ (relative to the point $z=\infty$). Let $\mathbb P_n:=\mathbb C_n[z]$ be the set of all algebraic polynomials with complex coefficients of degree $\leqslant\! {n}$. For an arbitrary polynomial $Q\in \mathbb C[z]$, $Q\not\equiv0$, let $\chi(Q)$ denote the zero counting measure of $Q$: $\chi(Q):=\sum_{\zeta\colon Q(\zeta)=0}\delta_\zeta$, where each zero of $Q$ is counted with multiplicity. Given a germ $f_\infty\!\in\!\mathscr H(\infty)$ and an arbitrary $n$, the Padé polynomials ${P_n,Q_n\!\in\!\mathbb P_n}$, $Q_n\not\equiv0$, are (not uniquely) defined by the following relation:
$$
\begin{equation}
R_n(z):=(Q_nf_\infty-P_n)(z)=O\biggl(\frac1{z^{n+1}}\biggr), \qquad z\to\infty;
\end{equation}
\tag{2}
$$
$R_n(z)$ is the so-called error function, and $[n/n]_{f_\infty}:=P_n/Q_n$ is the $n$th diagonal Padé approximant to $f_\infty$. The following result holds (see [17], [18] and also [1]). Theorem. Let $(f,f_\infty)\in\mathscr F$. Then, as $n\to\infty$, and Convergence $\xrightarrow{*}$ is understood as weak-$^*$ convergence in the space of measures. Convergence $\xrightarrow{\mathrm{cap}}$ in the interior of a domain $D$ means convergence in capacity on compact subsets of $D$ (see § 2.5). Note that in the case when $(f,f_\infty)\in\mathscr F$, the existence of an $S$-compact set for a germ $f_\infty$ follows already from the paper [11], treating a more special case than Stahl’s theory does. The proof of the above theorem has the following consequence (see inequality (18)). Corollary. If $(f,f_\infty) \in \mathscr F$ then the sequences $\{n-\operatorname{deg}{Q_n}\}$ and $\{n-\operatorname{deg}{P_n}\}$ are bounded. This result means that one of Gonchar’s conjectures relating to Padé approximations (see [1], Ch. 1, § 6, Conjecture 7) holds in the class $\mathscr F$. AcknowledgementThe author is sincerely grateful to the referee, whose comments contributed to correcting some gaps in proofs and improving the presentation of results. § 2. Proof of the theorem2.1Given two positive sequences $\{\alpha_n\}$ and $\{\beta_n\}$, the relation $\alpha_n\asymp \beta_n$ means that $0<C_1\leqslant \alpha_n/\beta_n\leqslant C_2<\infty$ for $n=1,2,\dots$ and some constants $C_1$ and $C_2$ independent of $n$. Given two sequences $\{\alpha_n(z)\}$ and $\{\beta_n(z)\}$ of holomorphic functions in a domain $\Omega$, the relation $\alpha_n\asymp\beta_n$ means that for each compact set $K\subset\Omega$ and all $n=1,2,\dots$ the inequality $0<C_1\leqslant |\alpha_n(z)/\beta_n(z)|\leqslant C_2<\infty$ holds for $z\in K$, where the constants $C_1$ and $C_2$ depend on $K$, but not on $n$. For such pairs of sequences (of numbers or functions) we obviously have $|\alpha_n/\beta_n|^{1/n}\to1$ as ${n\to\infty}$. Since $(f,f_\infty)\in\mathscr F$, we have $S=S(f)=\bigsqcup_{j=1}^p S_j$, where $S_j=\operatorname{arc}(a_{2j-1},a_{2j})$. Set $w^2=\prod_{j=1}^p(z-a_{2j-1})(z-a_{2j})$. Then the two-sheeted hyperelliptic Riemann surface $\mathfrak R_2(w)$ of the function $w$ is the Riemann surface associated with ${f_\infty\in\mathscr H(\infty)}$ and $f\in\mathscr F$ in accordance with Stahl’s theory. A point $\mathbf z$ on $\mathfrak R_2(w)$ is given by ${\mathbf z=(z,w)}$. The Riemann surface $\mathfrak R_2(w)$ can be considered as a two-sheeted covering of the Riemann sphere $\widehat{\mathbb C}$ with the canonical projection $\pi\colon\mathfrak R_2(w)\to\widehat{\mathbb C}$ defined by ${\pi(\mathbf z)=z}$. Let $\boldsymbol\Gamma:=\pi^{-1}(S)$. Then $\boldsymbol\Gamma$ partitions $\mathfrak R_2(w)$ into two domains. We call them the (open) sheets of $\mathfrak R_2(w)$. The function $w$ is single valued on this Riemann surface and takes opposite values on the sheets. We denote by $\mathfrak R_2^{(0)}$ the sheet of $\mathfrak R_2(w)$ where $w(z)/z^{p}\to1$ as $z\to\infty$ and refer to it as the zeroth sheet. The other sheet is denoted by $\mathfrak R^{(1)}_2$ and referred to as the first sheet of $\mathfrak R_2(w)$. Thus, $\mathfrak R_2(w)=\mathfrak R_2^{(0)}\sqcup\boldsymbol\Gamma\sqcup\mathfrak R_2^{(1)}$. Points on the two sheets of $\mathfrak R_2(w)$ are denoted by $z^{(j)}$, $j=0,1$. Clearly, $\pi(\mathfrak R_2^{(j)})=D$. Following tradition, we identify $\mathfrak R_2^{(0)}$ with the Stahl domain $D=\widehat{\mathbb C}\setminus{S}$ and $\infty^{(0)}$ with $\infty$, and we consider $f_\infty$ as a germ $f_{\infty^{(0)}}$ defined on $\mathfrak R_2(w)$. In general, $\mathfrak R_2(w)$ is not the Riemann surface of $f$, so that $f$ is not single valued on $\mathfrak R_2(w)$. But since $(f,f_\infty)\in\mathscr F$, the germ $f_{\infty^{(0)}}$ can be extended as a single-valued function from the point $\infty^{(0)}$ at infinity to the whole of the zero sheet $\mathfrak R^{(0)}_2$ and even farther, to a neighbourhood $V^{(0,1)}$ of the compact set $\boldsymbol\Gamma$ such that $V^{(0,1)}\cap\mathfrak R_2^{(1)}\neq\varnothing$ (in what follows we assume that this neighbourhood is sufficiently small). Since $\pi(\boldsymbol\Gamma)=S$ is Stahl’s compact set, the Green’s function $g_S(z,\infty)$ of $D$ can be lifted to $\mathfrak R_2(w)$ as a function $g(\mathbf z)$ of $\mathbf z$ with the following properties:
$$
\begin{equation*}
g(z^{(0)})=g_S(z,\infty)\quad\text{and} \quad g(z^{(1)})=-g(z^{(0)})<0.
\end{equation*}
\notag
$$
In what follows we assume that the open set $V^{(0,1)}$ satisfies
$$
\begin{equation*}
\partial V^{(0,1)}\cap\mathfrak R_2^{(1)}=\{z^{(1)}\colon g_S(z,\infty)=\log{R}\}, \qquad R>1.
\end{equation*}
\notag
$$
Moreover, $R>1$ is such that the number of connected components of the set $\{z^{(1)}\colon g_S(z,\infty)= \log{R}\}$ is equal to that of $S$. Let $\mathfrak D:=\mathfrak R_2^{(0)}\cup V^{(0,1)}$ be a domain on $\mathfrak R_2(w)$, and assume that $R$ is such that $f_{\infty^{(0)}}$ extends to a (single-valued) meromorphic function $f(\mathbf z)$ on $\mathfrak D$, $f\in\mathscr M(\mathfrak D)$. Then $R_n(z)$ is also extended to $\mathfrak D$ as a meromorphic function $R_n(\mathbf z)$. Let $q_m(z)= z^m+\dotsb$ be the polynomial whose zeros coincide (with multiplicities) with the projections of poles of $f(\mathbf z)$ that lie in $\mathfrak D$, and such that, moreover, the functions $\widetilde{f}:=q_mf$ and $q_mR_n$ are holomorphic in $\mathfrak D$. In what follows we assume that $n>m$. Finally, we assume without loss of generality that $S\ni0$. This is a purely technical condition: it ensures that the function $g_S(z,\infty)\mkern-1mu-\mkern-1mu\log|z|$ is continuous in any domain $\{z\colon\mkern-1mu g_S(z,\infty)\mkern-1mu>\mkern-1mu\log\rho\}$, $\rho\mkern-1mu>\mkern-1mu1$ (see (35)). Following tradition, below we identify the sheet $\mathfrak R_2^{(0)}\,{=}\,\mathfrak R_2^{(0)}(w)$ of $\mathfrak R_2(w)$ with the ‘physical’ extended complex plane $\widehat{\mathbb C}$ cut along the arcs forming Stahl’s $S$-compact set. We use the above notation in Figure 1. 2.2Given an arbitrary $\rho\in(1,R)$, where $R$ is as above, we denote by $\Gamma^{(1)}_\rho$ the set of points $z^{(1)}$ such that $g_S(z,\infty)=\log\rho$ for $z^{(1)}\in\Gamma^{(1)}_\rho$. Clearly, $g(z^{(1)})=-\log\rho$ for $z^{(1)}\in\Gamma^{(1)}_\rho$. The set $\Gamma^{(0)}_\rho$ is defined similarly: on it we have $g(z^{(0)})=\log\rho$. Set $\Gamma_\rho:=\pi(\Gamma^{(0)}_\rho)=\pi(\Gamma^{(1)}_\rho)$. For $\rho\in(1,R)$ let $D^{(1)}_\rho$ be the subdomain of $\mathfrak D$ with boundary $\partial D^{(1)}_\rho=\Gamma^{(1)}_\rho$ such that $\infty^{(0)}\in D^{(1)}_\rho$. In a similar way $D^{(0)}_\rho\subset\mathfrak D$, $\partial D^{(0)}_\rho=\Gamma^{(0)}_\rho$ and $\infty^{(0)}\in D^{(0)}_\rho$. Set
$$
\begin{equation}
u_n(\mathbf z):=\log|q_m(z)R_n(\mathbf z)|+(n+1-m)g(\mathbf z), \qquad \mathbf z\in\mathfrak D.
\end{equation}
\tag{5}
$$
The function $u_n$ is subharmonic in $\mathfrak D$. Hence by the maximum principle for subharmonic functions, for each $\rho\in(1,R)$ we have
$$
\begin{equation}
u_n(\mathbf z)\leqslant\max_{\boldsymbol\zeta\in\Gamma^{(1)}_\rho}u_n(\boldsymbol\zeta), \qquad \mathbf z\in D^{(1)}_\rho.
\end{equation}
\tag{6}
$$
It follows directly from (5) and (6) that for $\mathbf z\in\Gamma_\rho^{(0)}$ and $\boldsymbol\zeta\in\Gamma_\rho^{(1)}$ we have
$$
\begin{equation*}
\rho^{n+1-m} \bigl|R_n(z^{(0)})q_m(z)\bigr|\leqslant\frac1{\rho^{n+1-m}} \max_{\zeta\in\Gamma_\rho}\bigl|R_n(\zeta^{(1)})q_m(\zeta)\bigr|.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation}
\bigl|R_n(z^{(0)})q_m(z)\bigr|\leqslant\frac1{\rho^{2n+2-2m}} M_{n,1}(\rho), \qquad z\in\Gamma_\rho,
\end{equation}
\tag{7}
$$
where we set
$$
\begin{equation*}
M_{n,1}(\rho):=\max_{z\in\Gamma_\rho}|R_n(z^{(1)})q_m(z)|.
\end{equation*}
\notag
$$
Relations (5) and (6) imply that for $1<\rho_1<\rho_2<R$ we have
$$
\begin{equation}
M_{n,1}(\rho_1)\leqslant\biggl(\frac{\rho_1}{\rho_2}\biggr)^{n-m+1}M_{n,1}(\rho_2).
\end{equation}
\tag{8}
$$
It is easy to see that the following identity holds true:
$$
\begin{equation}
\begin{aligned} \, \notag R_n(z^{(0)}) &=Q_n(z)f(z^{(0)})-P_n(z) \\ &=Q_n(z)f(z^{(1)})-P_n(z)+Q_n(z)[f(z^{(0)})-f(z^{(1)})] \notag \\ &=R_n(z^{(1)})+Q_n(z)[f(z^{(0)})-f(z^{(1)})], \qquad z\in \Gamma_\rho. \end{aligned}
\end{equation}
\tag{9}
$$
From (7) and (8) it follows that
$$
\begin{equation}
\max_{z\in \Gamma_\rho}|Q_n(z)[f(z^{(0)})-f(z^{(1)})]q_m(z)| =M_{n,1}(\rho)(1+\varepsilon_n), \qquad \varepsilon_n\to0.
\end{equation}
\tag{10}
$$
Below in this section we consider only $\rho\in (1,R)$ such that $|q_m(z)[f(z^{(0)})-f(z^{(1)})]|\geqslant C(\rho)>0$ for $z\in\Gamma_\rho$ (clearly, $f(z^{(0)})-f(z^{(1)})\not\equiv0$). Set Then it follows from (10) that Since $\operatorname{deg}{Q_n}\leqslant{n}$, the Bernstein-Walsh theorem gives us the inequality
$$
\begin{equation}
|Q_n(z)|\leqslant e^{n g_{\Gamma_{\rho_1}}(z,\infty)}m_n(\rho_1), \qquad z\in\Gamma_{\rho_2}, \quad \rho_2>\rho_1,
\end{equation}
\tag{13}
$$
where $g_{\Gamma_{\rho_1}}(z,\infty)$ is the Green’s function for the domain $g_S(z,\infty)>\log\rho_1$. Clearly, $g_{\Gamma_{\rho_1}}(z,\infty)=g_S(z,\infty)-\log\rho_1$. From (13) we obtain the estimate
$$
\begin{equation}
m_n(\rho_2)\leqslant \biggl(\frac{\rho_2}{\rho_1}\biggr)^{n}m_n(\rho_1).
\end{equation}
\tag{14}
$$
Combining the relations (8), (12) and (14) we obtain
$$
\begin{equation}
m_n(\rho_2)\asymp \biggl(\frac{\rho_2}{\rho_1}\biggr)^n m_n(\rho_1)\quad\text{and} \quad M_{n,1}(\rho_2)\asymp \biggl(\frac{\rho_2}{\rho_1}\biggr)^n M_{n,1}(\rho_1).
\end{equation}
\tag{15}
$$
Let $Q_n(z)=z^{k_n}+\dotsb$, where $k_n=\operatorname{deg}{Q_n}\leqslant{n}$. Using the Bernstein-Walsh theorem again gives us the inequality
$$
\begin{equation}
|Q_n(z)|\leqslant e^{k_n g_{\Gamma_{\rho_1}}(z,\infty)}m_n(\rho_1), \qquad z\in\Gamma_{\rho_2}, \quad \rho_2>\rho_1.
\end{equation}
\tag{16}
$$
It follows from this relation that
$$
\begin{equation}
m_n(\rho_2)\leqslant\biggl(\frac{\rho_2}{\rho_1}\biggr)^{k_n}m_n(\rho_1).
\end{equation}
\tag{17}
$$
However, in accordance with (15) $m_n(\rho_2)\asymp (\rho_2/\rho_1)^n m_n(\rho_1)$. Hence from (17) we obtain the inequality
$$
\begin{equation}
\biggl(\frac{\rho_2}{\rho_1}\biggr)^{n}\leqslant C\biggl(\frac{\rho_2}{\rho_1}\biggr)^{k_n},
\end{equation}
\tag{18}
$$
where $C=C(\rho_1,\rho_2)$, $k_n\leqslant{n}$ and $1<\rho_1<\rho_2$. It follows directly from (18) that
$$
\begin{equation}
\frac{\operatorname{deg}{Q_n}}{n}=\frac{k_n}{n}\to1, \qquad n\to\infty.
\end{equation}
\tag{19}
$$
Note that relations (15) are ‘comparisonal’, that is, they are preserved by renormalizations of the polynomial $Q_n$ replacing $Q_n(z)=z^{k_n}+\dotsb$ by $c_nQ_n$, $c_n\neq0$. In what follows we fix a certain analogue of the so-called spherical normalization for $Q_n$ (see [5] and [3]). 2.3Let $D_\rho:=\{z\in\widehat{\mathbb C}\colon g_S(z,\infty)>\log\rho\}$, $\rho>1$, and let $g_{\Gamma_\rho}(z,\infty)$ be the Green’s function for $D_\rho$. Then
$$
\begin{equation*}
g_{\Gamma_\rho}(z,\infty)=g_S(z,\infty)-\log\rho =\log{|z|}+\gamma_\rho+o(1)=\log|z|+\gamma_S-\log\rho+o(1).
\end{equation*}
\notag
$$
Thus, $\gamma_\rho=\gamma_S-\log\rho$. Set
$$
\begin{equation}
\widetilde{u}_n(z):=\frac1{k_n}\log|Q_n(z)|-g_{\Gamma_\rho}(z,\infty)-\frac1{k_n}\log m_n(\rho).
\end{equation}
\tag{20}
$$
Since the function $\widetilde{u}_n$ is subharmonic in $D_\rho$ and $\widetilde{u}_n\leqslant0$ on $\Gamma_\rho$, it follows that $\widetilde{u}_n\leqslant 0$ in $D_\rho$ and $\widetilde{u}_n(\infty)\leqslant0$. Thus we have $\log\rho-\gamma_S\leqslant\log m_n(\rho)^{1/k_n}$, and finally we obtain
$$
\begin{equation}
m_n(\rho)^{1/k_n}\geqslant \rho e^{-\gamma_S}=\rho\operatorname{cap}(S).
\end{equation}
\tag{21}
$$
For $z\in K\Subset D_\rho$ and $\zeta\in\Gamma_\rho$ we have $g_{\Gamma_\rho}(\zeta,z)=0$ by definition. Using the symmetry of the Green’s function in its arguments we obtain $g_{\Gamma_\rho}(z,\zeta)=0$ for $z\in K\Subset D_\rho$ and $\zeta\in\Gamma_\rho$. Now let us extend $g_{\Gamma_\rho}(z,\zeta)$ by identical zero inside $\Gamma_\rho$ with respect to $\zeta$: $g_{\Gamma_\rho}(z,\zeta)\equiv0$ for $z\in D_\rho$ and $\zeta \in\operatorname{int} \Gamma_\rho$. Throughout the rest of this section we consider only $\rho>1$ such that $Q_n(z)\neq0$ for $z\in\Gamma_\rho$ and all $n\in\mathbb N$. Let $Q_n(z)=\prod_{j=1}^{k_n}(z-\zeta_{n,j})$. Set
$$
\begin{equation}
v_n(z):=\frac1{k_n}\log|Q_n(z)|-g_{\Gamma_\rho}(z,\infty)+\frac1{k_n}\sum_{j=1}^{k_n} g_{\Gamma_\rho}(z,\zeta_{n,j})-\frac1{k_n}\log m_n(\rho),
\end{equation}
\tag{22}
$$
for $z\in D_{\rho}$. Then $v_n$ is a harmonic function in $D_\rho$ and $v_n\leqslant0$ on $\Gamma_\rho$. From this it follows that for $z\in \Gamma_{\rho_2}$, where $\rho<\rho_2<R$, we have
$$
\begin{equation}
|Q_n(z)|\exp\biggl\{\sum_{j=1}^{k_n} g_{\Gamma_\rho}(z,\zeta_{n,j})\biggr\} \leqslant m_n(\rho) \exp\bigl\{k_ng_{\Gamma_\rho}(z,\infty) \bigr\}.
\end{equation}
\tag{23}
$$
Let $z_n^*\in \Gamma_{\rho_2}$ be such that $|Q_n(z_n^*)|=m_n(\rho_2)$. Then from (23) we obtain
$$
\begin{equation}
m_n(\rho_2)\exp\biggl\{\sum_{j=1}^{k_n}g_{\Gamma_\rho}(z_n^*,\zeta_{n,j}) \biggr\}\leqslant m_n(\rho)\biggl(\frac{\rho_2}{\rho}\biggr)^{k_n}.
\end{equation}
\tag{24}
$$
Finally, from (24) and (15) we see that
$$
\begin{equation}
\exp\biggl\{\frac1{k_n}\sum_{j=1}^{k_n}g_{\Gamma_\rho}(z_n^*,\zeta_{n,j}) \biggr\}\leqslant C(\rho,\rho_2)^{1/k_n}.
\end{equation}
\tag{25}
$$
Now let the probability measure $\mu$ be a limit point of the sequence $\{\mu_n\}$, where $\mu_n=\dfrac1{k_n}\chi(Q_n)$, that is, let $\mu_n\xrightarrow{\ast}\mu$ as $n\to\infty$, $n\in\Lambda\subset\mathbb N$. We also assume that $z_n^*\to z^*\in \Gamma_{\rho_2}$ as $n\to\infty$, $n\in\Lambda$. Then by the descendence principle (see [7], Ch. I, § 3, Theorem 1.3; [14], Ch. I, Theorem 6.8; [4] and also the lemma below) we have
$$
\begin{equation}
\int g_{\Gamma_\rho}(z^*,\zeta)\,d\mu^{(\rho)}(\zeta)\leqslant 0, \qquad z^*\in\Gamma_{\rho_2}, \quad \rho_2>\rho,
\end{equation}
\tag{26}
$$
where $\mu^{(\rho)}\!=\!\mu|_{\overline{D}_\rho}$. It follows directly from (26) that $\mu|_{\overline{D}_\rho}\!\!=\!0$, so that ${\operatorname{supp}{\mu}\!\subset\!\widehat{\mathbb C}\!\setminus\! D_\rho}$. Since this is true for all $\rho>1$ except a countable set of values of $\rho$, we have $\operatorname{supp}{\mu}\subset S$. 2.4Thus we have obtained that $k_n/n\to1$ and each limit point $\mu$ of the sequence $\biggl\{\dfrac1{k_n}\chi(Q_n)\biggr\}$ satisfies the condition $\operatorname{supp}\mu\subset S$. Now we show that $\mu=\lambda_S$. Fix some $\rho>1$ and set
$$
\begin{equation}
Q^{*}_n(z):=\prod_{\zeta_{n,j}\notin D_\rho}(z-\zeta_{n,j})\cdot \prod_{\zeta_{n,j}\in D_\rho}\biggl(1-\frac{z}{\zeta_{n,j}}\biggr).
\end{equation}
\tag{27}
$$
We denote the error function corresponding to this normalization of the Padé polynomial by $R^{*}_n$. Let $m^{*}_n(\rho')$ and $M^{*}_{n,1}(\rho')$, $\rho'>1$, be the analogues of the quantities $m_n(\rho')$ for $Q_n$ and $M_{n,1}(\rho')$ for $R_n$, which are obtained by replacing $Q_n$ by $Q^{*}_n$ and $R_n$ by $R^{*}_n$. Then (15) also holds for these quantities. Set1
$$
\begin{equation*}
u_n(z):=\frac1{k_n}\log\frac1{|Q^{*}_n(z)|}-\frac1{k_n}\sum_{j=1}^{k_n} g_{\Gamma_\rho}(z,\zeta_{n,j})+g_{\Gamma_\rho}(z,\infty), \qquad z\in D_\rho.
\end{equation*}
\notag
$$
Then $\{u_n\}$ is a sequence of harmonic functions in $D_\rho$ such that is a harmonic function in $D_\rho$ with the following properties (see (15)):
$$
\begin{equation}
\begin{gathered} \, u(z)=V^{\mu}(z)-V^{\lambda_S}(z)+\mathrm{const}, \\ \min_{z\in\Gamma_{\rho_1}}u(z)=\min_{z\in\Gamma_{\rho_2}}u(z) \quad\text{for all } \rho_1,\rho_2>\rho. \end{gathered}
\end{equation}
\tag{28}
$$
In fact, let $\chi(Q_n)=\mu_{n,1}+\mu_{n,2}$, where
$$
\begin{equation*}
\mu_{n,1}=\sum_{\zeta_{n,j}\notin D_\rho}\delta_{\zeta_{n,j}}\quad\text{and} \quad \mu_{n,2}=\sum_{\zeta_{n,j}\in D_\rho}\delta_{\zeta_{n,j}}.
\end{equation*}
\notag
$$
Then by the above, as $n\to\infty$, $n\in\Lambda$, we have
$$
\begin{equation*}
\frac1n\mu_{n,1}\xrightarrow{*}\mu\quad\text{and} \quad \frac1n\mu_{n,2}\xrightarrow{*}0.
\end{equation*}
\notag
$$
Since $g_{\Gamma_\rho}(z,\zeta)\equiv0$ for $\zeta\notin D_\rho$, it follows that
$$
\begin{equation}
\sum_{j=1}^{k_n}g_{\Gamma_\rho}(z,\zeta_{n,j})= \int g_{\Gamma_\rho}(z,\zeta)\,d\mu_{n,2}(\zeta).
\end{equation}
\tag{29}
$$
We have
$$
\begin{equation}
\begin{aligned} \, \notag \log\frac1{|Q^{*}_n(z)|} &=\log\frac1{|Q_n(z)|}+\prod_{\zeta_{n,j}\in D_\rho}\log|\zeta_{n,j}| \\ &=V^{\mu_{n,1}}(z)+V^{\mu_{n,2}}(z)+\int\log|\zeta|\,d\mu_{n,2}(\zeta). \end{aligned}
\end{equation}
\tag{30}
$$
Let $\widetilde{\mu}_{n,2}$ be the balayage of the measure $\mu_{n,2}$ from $D_\rho$ to $\Gamma_\rho$. Then
$$
\begin{equation}
V^{\mu_{n,2}}(z)-\int g_{\Gamma_\rho}(z,\zeta)\,d\mu_{n,2}(\zeta) =V^{\widetilde{\mu}_{n,2}}(z)+c_n,
\end{equation}
\tag{31}
$$
where
$$
\begin{equation}
c_n=-\int g_{\Gamma_\rho}(\zeta,\infty)\,d\mu_{n,2}(\zeta).
\end{equation}
\tag{32}
$$
We conclude from (29)–(32) that for $z\in D_\rho$
$$
\begin{equation}
\begin{aligned} \, \notag u_n(z) &=\frac1{k_n}V^{\mu_{n,1}}(z)+\frac1{k_n}V^{\widetilde{\mu}_{n,2}}(z) \\ &\qquad +\frac1{k_n}\int (\log|\zeta|-g_{\Gamma_\rho}(\zeta,\infty))\,d\mu_{n,2}(\zeta)+g_{\Gamma_\rho}(z,\infty). \end{aligned}
\end{equation}
\tag{33}
$$
Since $k_n/n\to1$ and $\operatorname{supp}{\widetilde{\mu}_{n,2}}, \operatorname{supp}{\mu_{n,1}} \subset \widehat{\mathbb C}\setminus D_\rho$, we see that, as $n\to\infty$, $n\in\Lambda$, we have
$$
\begin{equation}
\frac1{k_n}V^{\mu_{n,1}}(z)\to V^{\mu}(z), \qquad \frac1{k_n}V^{\widetilde{\mu}_{n,2}}(z)\to 0
\end{equation}
\tag{34}
$$
and
$$
\begin{equation}
\frac1{k_n} \int \bigl(\log|\zeta|-g_{\Gamma_\rho}(\zeta,\infty)\bigr)\,d\mu_{n,2}(\zeta) \to0
\end{equation}
\tag{35}
$$
uniformly on compact subsets of $D_\rho$. In turn, from (15) and (31)–(35) we obtain (28). It follows from (28) that $u(z)=\mathrm{const}$, so that $V^\mu(z)=V^{\lambda_S}(z)$ for $z\in D$. Since $S$ contains no interior points, we finally obtain ${\mu=\lambda_S}$. Thus the equilibrium measure $\lambda_S$ is the unique limit point of the sequence $\dfrac1n\chi(Q_n)$. Thus we have shown that the following limits exist for any $\rho>1$:
$$
\begin{equation}
\lim_{n\to\infty}m^{*}_n(\rho)^{1/n}=\lim_{n\to\infty}M^{*}_{n,1}(\rho)^{1/n}=\rho \operatorname{cap}(S), \qquad \rho>1,
\end{equation}
\tag{36}
$$
$$
\begin{equation}
\lim_{n\to\infty}\max_{z\in S}|Q^{*}_n(z)|^{1/n}=\operatorname{cap}(S)=e^{-\gamma_S}\quad\text{and} \quad \lim_{n\to\infty}\max_{z\in\Gamma_\rho}|R^{*}_n(z^{(0)})|^{1/n}=\frac1\rho e^{-\gamma_S}.
\end{equation}
\tag{37}
$$
It is clear that the above relations do not depend on the choice of $\rho>1$ in the $*$-normalization (27) for $Q_n$. Thus, we can take the $*$-normalization with respect to $R>1$ which we fixed in § 2.2. We stick to this convention in what follows. 2.5It is easy to deduce from the above that
$$
\begin{equation}
|q_m(z)R^{*}_n(z)|^{1/n}\xrightarrow{\mathrm{cap}} \operatorname{cap}(S) e^{-g_S(z,\infty)}, \qquad n\to\infty.
\end{equation}
\tag{38}
$$
Indeed, let
$$
\begin{equation}
\widetilde{v}_n(z):=\frac1{n-m+1}\log|q_m(z)R^{*}_n(z)|+g_S(z,\infty)+\gamma_S.
\end{equation}
\tag{39}
$$
Then for each $\rho>1$ the function $\widetilde{v}_n$ is subharmonic in $D_\rho$, and it follows from (37) that for $z\in\Gamma_\rho$, where $C_n\to0$ as $n\to\infty$. From (40) we obtain
$$
\begin{equation}
|q_m(z)R^{*}_m(z)|^{1/(n-m+1)}\leqslant\operatorname{cap}(S)e^{-g_S(z,\infty)}e^{C_n}
\end{equation}
\tag{41}
$$
for $z\in D_\rho$. Thus,
$$
\begin{equation}
\varlimsup_{n\to\infty}|q_m(z)R^{*}_n(z)|^{1/n}\leqslant\operatorname{cap}(S)e^{-g_S(z,\infty)}.
\end{equation}
\tag{42}
$$
Relation (38) follows from (37) and (42) by the two-constants theorem (cf. [15], § 3.8, formulae (31)–(36), and [3]). In fact, (38) means that for an arbitrary compact set $K\subset D=\widehat{\mathbb C}\setminus{S}$ and each $\varepsilon>0$ we have (cf. [15], § 3.8, formulae (31)–(36), and [19], Theorem 1.1)
$$
\begin{equation*}
\operatorname{cap}\bigl(K_{1,n}(\varepsilon)\cup K_{2,n}(\varepsilon)\bigr)\to0, \qquad n\to\infty,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
K_{1,n}(\varepsilon):=\bigl\{z\in K\colon |q_m(z)R^{*}_n(z)|^{1/n}\geqslant \operatorname{cap}(S)e^{-g_S(z,\infty)+2\varepsilon}\bigr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
K_{2,n}(\varepsilon):=\bigl\{z\in K\colon |q_m(z)R^{*}_n(z)|^{1/n}\leqslant \operatorname{cap}(S)e^{-g_S(z,\infty)-2\varepsilon}\bigr\}.
\end{equation*}
\notag
$$
It follows from (42) that we must consider only the case when $K_n(\varepsilon):=K_{2,n}(\varepsilon)$, that is, we must show that $\operatorname{cap}(K_n(\varepsilon))\to0$ as $n\to\infty$. Let $\rho_1$ and $\rho_2$, $1<\rho<\rho_1<\rho_2$, be numbers such that $K\subset G:=D_{\rho_2}\setminus\overline{D}_{\rho_1}$. Then $K_n(\varepsilon)\subset G$ for all $n\in\mathbb N$. Throughout the rest of this subsection we consider only compact sets $K$ lying in the open set $G$. Then the properties of the logarithmic capacity $\operatorname{cap}(K)$ of $K$ are equivalent to the properties of the Green’s capacity $\operatorname{cap}_\rho(K)$ of $K$ with respect to the compact set $\Gamma_\rho=\partial D_\rho$ in the following sense: the relation $\operatorname{cap}_\rho(K_n)\to0$ as $n\to\infty$ is equivalent to $\operatorname{cap}(K_n)\to0$ (see [7], [14], [4] and formula (49) below). Thus, we must prove that $\operatorname{cap}_\rho(K_n(\varepsilon))\to0$ as $n\to\infty$. Assume the contrary: let $\operatorname{cap}_\rho(K_n(\varepsilon))\geqslant\delta$ for some $\delta>0$ and $n\in\Lambda$, $n\to\infty$. Since $q_mR^{*}_n$ is a holomorphic function in $D_\rho\supset K_n(\varepsilon)$, each point $z\in K_n(\varepsilon)$ has a neighbourhood $U(z)$ such that
$$
\begin{equation*}
|q_m(\zeta)R^{*}_n(\zeta)|^{1/n}\leqslant \operatorname{cap}(S)e^{-g_S(\zeta,\infty)-\varepsilon}, \qquad \zeta\in U(z), \quad z\in K_n(\varepsilon).
\end{equation*}
\notag
$$
Hence there exists a compact set $F_n(\varepsilon)=\bigcup_{j=1}^N \overline{U}(z_j)$ such that $F_n(z)\supset K_n(\varepsilon)$, $F_n(\varepsilon)\subset G$, $F_n(\varepsilon)$ is a regular compact set, $\operatorname{cap}_\rho(F_n(\varepsilon))\geqslant\delta>0$ for $n\in\Lambda$ and
$$
\begin{equation}
|q_m(z)R^{*}_n(z)|^{1/(n-m+1)}\leqslant \operatorname{cap}(S)e^{-g_S(z,\infty)-\varepsilon}, \qquad z\in F_n(\varepsilon), \quad n\in\Lambda.
\end{equation}
\tag{43}
$$
Since $q_m(z)R^{*}_n(z)$ is a holomorphic function in $G\supset K$, by the maximum principle inequality (43) also holds in the polynomial hull of $F_n(\varepsilon)$. Thus we can assume that $F_n(\varepsilon)$ does not separate the complex plane. Set $D_n(\varepsilon):=D_\rho\setminus F_n(\varepsilon)$. Then $D_n(\varepsilon)$ is a domain with boundary $\partial D_n(\varepsilon)=\Gamma_\rho\cup\partial F_n(\varepsilon)$. Let $\omega_n(z)$ be the harmonic measure of $\partial F_n(\varepsilon)$ with respect to $\Gamma_\rho$, that is, let $\omega_n(z)$ be the harmonic function in $D_n(\varepsilon)$ that is continuous in the closure of $\overline{D}_\rho(\varepsilon)$ and satisfies $\omega_n(z)\equiv 0$ for $z\in\Gamma_\rho$ and $\omega_n(z)\equiv1$ for $z\in\partial F_n(\varepsilon)$. Set
$$
\begin{equation}
\begin{aligned} \, w_n(z) &:=\frac1{n-m+1}\log|q_m(z)R^{*}_n(z)| \nonumber \\ &\qquad +g_S(z,\infty)+\gamma_S-\eta(1-\omega_n(z))+\varepsilon\omega_n(z), \end{aligned}
\end{equation}
\tag{44}
$$
where $\eta>0$ is an arbitrary positive number. It follows from (42) and (43) that for ${n\in\Lambda}$, $n\geqslant n_0(\eta)$, we have Fix $\rho_3>\rho_2$. From (44) and (45), for $n\geqslant n_0$ we obtain
$$
\begin{equation}
|q_m(z)R^{*}_n(z)|^{1/(n-m+1)}\leqslant \frac1{\rho_3}e^{-\gamma_S}e^{\eta(1-\omega_n(z))-\varepsilon\omega_n(z)}
\end{equation}
\tag{46}
$$
uniformly for $z\in\Gamma_{\rho_3}$. For an arbitrary compact set $K\subset G$ with positive capacity $\operatorname{cap}(K)$ and for an arbitrary unit measure $\mu$ with support in $K$, $\operatorname{supp}(\mu)\subset K$, we define the Green’s potential of $\mu$ relative to the domain $D_\rho$ by Since $\operatorname{cap}(K)>0$, there exists a unique unit measure $\lambda_K$ with support on $K$ such that
$$
\begin{equation}
G_{\lambda_K}(z)=\mathrm{const}=\gamma_\rho(K) \quad\text{quasi-everywhere on } K.
\end{equation}
\tag{48}
$$
Since $\operatorname{cap}(K)>0$, the constant $\gamma_\rho(K)$ is finite, and therefore the quantity is positive. Since $F_n(\varepsilon)$ is a regular compact set and $\operatorname{cap}_\rho(F_n(\varepsilon))\geqslant\delta>0$, it follows that $G_{\lambda_{F_n(\varepsilon)}}(z)\equiv\gamma_\rho(F_n(\varepsilon))$ on $\partial F_n(\varepsilon)$ and $\gamma_\rho(F_n(\varepsilon))\leqslant\log(1/\delta)$ for $n\in\Lambda$. Hence the harmonic measure $\omega_n(z)$ introduced above has the representation
$$
\begin{equation}
\omega_n(z)=\frac1{\gamma_\rho(F_n(\varepsilon))}G_{\lambda_{F_n(\varepsilon)}}(z), \qquad z\in D_\rho(\varepsilon).
\end{equation}
\tag{50}
$$
Let
$$
\begin{equation*}
m:=\min_{z\in\Gamma_{\rho_3},\, \zeta\in\overline{G}} g_{D_\rho}(z,\zeta)>0.
\end{equation*}
\notag
$$
Since $ \lambda_{F_n(\varepsilon)}(1)=1$, it follows that
$$
\begin{equation}
\min_{z\in\Gamma_{\rho_3}}\omega_n(z)\geqslant\frac{m}{\gamma_\rho(F_n(\varepsilon))} \geqslant\frac{m}{\log(1/\delta)}=:m_0>0.
\end{equation}
\tag{51}
$$
It follows from (46) and (51) that, uniformly in $z\in D_{\rho_3}$, we have
$$
\begin{equation}
|q_m(z)R^{*}_n(z)|^{1/(n-m+1)}\leqslant \frac1{\rho_3}e^{-\gamma_S}e^{\eta-\varepsilon m_0}, \qquad z\in\Gamma_{\rho_3}.
\end{equation}
\tag{52}
$$
Hence for $\rho_3$ such that $q_m(z)\neq0$, on $\Gamma_{\rho_3}$ we have the relation
$$
\begin{equation}
\varlimsup_{n\to\infty}\max_{z\in\Gamma_{\rho_3}}|R^{*}_n(z)|^{1/n}\leqslant \frac1{\rho_3}e^{-\gamma_S}e^{\eta-\varepsilon m_0}, \qquad z\in\Gamma_{\rho_3},
\end{equation}
\tag{53}
$$
where $\varepsilon>0$, $m_0>0$ is fixed and $\eta>0$ is arbitrary. Now letting $\eta$ in (53) tend to zero we arrive at a contradiction to (37). Since $\dfrac1n\chi(Q_n)\to\lambda_S$, in the interior of $D$ we have
$$
\begin{equation}
|Q_n(z)|^{1/n}\xrightarrow{\mathrm{cap}} e^{-V^{\lambda_S}(z)}, \qquad n\to\infty.
\end{equation}
\tag{54}
$$
Relations (38) and (54) yield (4). The theorem is proved. § 3. Appendix3.1In fact, the class of admissible multivalued analytic functions for which the above proof is valid is much wider than the class $\mathscr F$ of algebraic functions satisfying conditions (I) and (II) in the above definition. It particular, our approach holds for a certain class of analytic functions produced by the inverse Joukowsky function. More precisely, let where $z\in\widehat{\mathbb C}\setminus\Delta$, $\Delta=[-1,1]$ and the branch of $(\,\cdot\,)^{1/2}$ is chosen so that $\varphi(z)/z\to2$ as $z\to\infty$. Let $1<A<B<\infty$ and let $a:=(A+1/A)/2$ and $b:=(B+1/B)/2$. Then the function
$$
\begin{equation}
\mathfrak f:= \mathfrak f(z;\Delta):=\biggl[\biggl(A-\frac1{\varphi(z)}\biggr)\biggl(B-\frac1{\varphi(z)}\biggr)\biggr]^{1/2}
\end{equation}
\tag{55}
$$
is an algebraic function of the fourth order with square root singularities. Let $\Sigma(\mathfrak f)=\{\pm1,a,b\}$ be the corresponding set of branch points of $\mathfrak f(z;\Delta)$. Under the above condition on $(\,\cdot\,)^{1/2}$ we have $\mathfrak f_\infty\in\mathscr H(\widehat{\mathbb C}\setminus\Delta)$, and Stahl’s compact set for $\mathfrak f_\infty$ is the interval $[-1,1]$: $S(\mathfrak f_\infty)=[-1,1]$. Now let $\varphi_{\Delta_j}(z)$ be the inverse Joukowsky function corresponding to an interval $\Delta_j:=[\alpha_j,\beta_j]$, $j=1,\dots,m$, where $\Delta_j\cap \Delta_k=\varnothing$, $j\neq k$. Set
$$
\begin{equation}
\mathfrak f(z):=\prod_{j=1}^m {\mathfrak f}(z;\Delta_j),
\end{equation}
\tag{56}
$$
where each $\mathfrak f(z;\Delta_j)$ is defined by (55) for $\varphi_{\Delta_j}$ in place of $\varphi$ and some $A_j$ and $B_j$ in place of $A$ and $B$. If all intervals $\Delta_j$ are real, $\Delta_j\subset\mathbb R$, $j=1,\dots,m$, then Stahl’s compact set has the following form: $S(\mathfrak f)=\bigsqcup_{j=1}^m\Delta_j$. Since all functions in $\mathbb C(z,\mathfrak f)$ satisfy condition (II) in the definition of $\mathscr F$, our approach also holds for the germ $f_\infty$ of an arbitrary function $f$ in the class $\mathbb C(z,\mathfrak f)$. Another nontrivial admissible class of multivalued analytic functions arises when we assume that at least one of the branch points $\alpha_j$ and $\beta_j$, $j=1,\dots,m$, does not belong to the real line. We can also generalize (55) in the following way. Set
$$
\begin{equation}
{\mathfrak f}(z;\Delta):=\prod\biggl(A-\frac1{\varphi(z)} \biggr)^\alpha\biggl(B-\frac1{\varphi(z)}\biggr)^\beta \dotsb\biggl(C-\frac1{\varphi(z)}\biggr)^\gamma,
\end{equation}
\tag{57}
$$
where $\alpha,\beta,\dots,\gamma\in\mathbb C\setminus\mathbb Z$ and $\alpha+\beta+\dots+\gamma\in\mathbb Z$. Then we obtain a class of functions $\mathbb C(z,\mathfrak f)$, which is not a subclass of $\mathscr F$, but for which the proof in § 2 is also valid. 3.2Let $D\subset\widehat{\mathbb C}$, $D\neq\widehat{\mathbb C}$, be a regular domain and $g_D(\zeta,z)$, $z,\zeta\in D$, be the Green’s function for $D$ with logarithmic singularity at $\zeta=z$. For an arbitrary (positive Borel) measure $\mu$ we define its Green potential $G_D(z;\mu)$ with respect to $D$ by
$$
\begin{equation*}
G_D(z;\mu):=\int g_D(\zeta,z)\,d\mu(\zeta), \qquad z\in D.
\end{equation*}
\notag
$$
Then the following result holds. Lemma. Let $K\subset D$ be a compact set, let $\{z_n\}_{n\in\mathbb N}\subset K$ be a sequence of points such that $z_n\to z^*$ as $n\to\infty$ and $\{\mu_n\}$ be a sequence of measures such that $\operatorname{supp}\mu_n\subset\overline{D}$, $\mu_n\to\mu$ as $n\to\infty$, where $\mu$ is a probability measure. Then Proof. Since $K\subset D$, for all $z\in K$ and each sufficiently small $\varepsilon\in(0,\varepsilon_0)$, ${\varepsilon_0\,{=}\,\varepsilon_0(K)}$, the level curve $L_\varepsilon:=\{\zeta\colon g_D(\zeta,z)= 1/\varepsilon\}$ is a closed analytic curve enclosing the point $z$ and such that $g_D(\zeta,z)> 1/\varepsilon$ inside $L_\varepsilon$ and $g_D(\zeta,z)<1/\varepsilon$ outside $L_\varepsilon$.
Let $D_\varepsilon\ni z$ be the domain with boundary $\partial D_\varepsilon=L_\varepsilon$. For $\varepsilon\in(0,\varepsilon_0)$ we set (cf. [7], Ch. I, § 3.7)
$$
\begin{equation*}
g^{(\varepsilon)}_D(\zeta,z):= \begin{cases} g_D(\zeta,z),& z\in D\setminus D_\varepsilon, \\ \dfrac1\varepsilon,& z\in D_\varepsilon. \end{cases}
\end{equation*}
\notag
$$
Also set
$$
\begin{equation*}
G^{(\varepsilon)}_D(z;\mu):=\int g^{(\varepsilon)}_D(\zeta,z)\,d\mu(\zeta)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
G^{(\varepsilon)}_D(z;\mu_n):=\int g^{(\varepsilon)}_D(\zeta,z)\,d\mu_n(\zeta), \qquad n=1,2,\dotsc\,.
\end{equation*}
\notag
$$
Then for any fixed $\varepsilon\in(0,\varepsilon_0)$ the family of functions $\{G^{(\varepsilon)}_D(z;\mu_n)\}_{n\in\mathbb N}$ is equicontinuous for $z\in K$. Moreover, for each fixed $z\in K$ the functions $g^{(\varepsilon)}_D(\zeta,z)$ and $G^{(\varepsilon)}_D(z;\mu)$ are monotonically increasing as $\varepsilon\to0$. Thus,
$$
\begin{equation}
G^{(\varepsilon)}_D(z^*,\mu)=\lim_{n\to\infty}G^{(\varepsilon)}_D(z_n;\mu_n) \leqslant \varliminf_{n\to\infty}G_D(z_n,\mu_n).
\end{equation}
\tag{59}
$$
From (59), letting $\varepsilon\to0$ by Beppo Levi’s monotone convergence theorem we obtain
$$
\begin{equation*}
G_D(z^*;\mu)\leqslant\varliminf_{n\to\infty}G_D(z_n;\mu_n).
\end{equation*}
\notag
$$
The proof is complete.
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\Bibitem{Sue22} |
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