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This article is cited in 5 scientific papers (total in 5 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 Note that a short proof of the existence of an S-compact set in this case was proposed by Rakhmanov (1994) in the unpublished paper [11] (also see [8]).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 F of admissible multivalued functions under consideration in our paper. For a multivalued function f we agree that f belongs to 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∞∈H(∞) 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∞)∈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∞∈H(∞), and let S=S(f∞) be Stahl’s S-compact set corresponding to the germ f∞. Let λS be the probability equilibrium measure with support on S and VλS(z) be the corresponding logarithmic potential of λS: where gS(z,∞) is the Green’s function for the Stahl domain D=ˆC∖S with logarithmic singularity at the point at infinity z=∞, and γS is the Robin constant for D (relative to the point z=∞).Let Pn:=Cn[z] be the set of all algebraic polynomials with complex coefficients of degree ⩽n. For an arbitrary polynomial Q∈C[z], Q≢0, let χ(Q) denote the zero counting measure of Q: χ(Q):=∑ζ:Q(ζ)=0δζ, where each zero of Q is counted with multiplicity. Given a germ f∞∈H(∞) and an arbitrary n, the Padé polynomials Pn,Qn∈Pn, Qn≢0, are (not uniquely) defined by the following relation: Rn(z) is the so-called error function, and [n/n]f∞:=Pn/Qn is the nth diagonal Padé approximant to f∞.The following result holds (see [17], [18] and also [1]). and Convergence ∗→ is understood as weak-∗ convergence in the space of measures. Convergence 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∞)∈F, the existence of an S-compact set for a germ f∞ 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)). This result means that one of Gonchar’s conjectures relating to Padé approximations (see [1], Ch. 1, § 6, Conjecture 7) holds in the class 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 {αn} and {βn}, the relation αn≍βn means that 0<C1⩽αn/βn⩽C2<∞ for n=1,2,… and some constants C1 and C2 independent of n. Given two sequences {αn(z)} and {βn(z)} of holomorphic functions in a domain Ω, the relation αn≍βn means that for each compact set K⊂Ω and all n=1,2,… the inequality 0<C1⩽|αn(z)/βn(z)|⩽C2<∞ holds for z∈K, where the constants C1 and C2 depend on K, but not on n. For such pairs of sequences (of numbers or functions) we obviously have |αn/βn|1/n→1 as n→∞. Since (f,f∞)∈F, we have S=S(f)=⨆pj=1Sj, where Sj=arc(a2j−1,a2j). Set w2=∏pj=1(z−a2j−1)(z−a2j). Then the two-sheeted hyperelliptic Riemann surface R2(w) of the function w is the Riemann surface associated with f∞∈H(∞) and f∈F in accordance with Stahl’s theory. A point z on R2(w) is given by z=(z,w). The Riemann surface R2(w) can be considered as a two-sheeted covering of the Riemann sphere ˆC with the canonical projection π:R2(w)→ˆC defined by π(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 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 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 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.
Citation in format AMSBIB
\Bibitem{Sue22} |
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