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This article is cited in 1 scientific paper (total in 1 paper) Scalar approaches to the limit distribution of the zeros of Hermite–Padé polynomials for a Nikishin system S. P. Suetin Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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    1. Introduction1.1.This paper surveys the results obtained by means of scalar approaches to the limit distribution of the zeros of Hermite–Padé polynomials for a pair of functions forming a Nikishin system. In fact, the text falls into three main parts. In the first part (§ 2) we discuss our results obtained on the basis of the first scalar approach proposed in [90] (2018). In the second part (§§ 3 and 4), using the second scalar approach, which is based on the maximum principle alone, we prove a result on the limit distribution of the zeros of Hermite–Padé polynomials which was stated in [97]. Finally, in the third part (§ 5) we compare our results with the ones due to Stahl [84] and obtained in the framework of an approach he proposed in 1987–1988. On the whole, all three approaches are based on potential theory on Riemann surfaces. Note that the scalar approaches under discussion here can naturally be regarded as an alternative to the classical vector approach of Gonchar–Rakhmanov, which was applied for the first time in [39] to an Angelesco system of functions and was subsequently generalized by Nikishin [63] to another system of functions, which is now called a Nikishin system. The first scalar approach, proposed in [90], was further developed in [91], [93], [94], and [45]. In particular, in [95], Theorem 1, we showed that for a pair of functions $f_1$, $f_2$ forming a classical Nikishin system this approach is equivalent to the vector approach of Gonchar and Rakhmanov. Recall that the traditional vector approach to the problem of the limit distribution of the zeros of Hermite–Padé polynomials in the case of a pair of functions forming a Nikishin system is based on a solution to a vector-valued potential-theoretic equilibrium problem stated in terms of a $ 2\times2 $-matrix (known now as a Nikishin matrix); see first of all [63], [64], [41], and [38]; also see [10], [18], [53], [55], [11] and the references in these papers. In §§ 3 and 4 we present another scalar approach, announced previously in [97] and [102] and based exclusively on the maximum principle for subharmonic functions. We stress that we do not use orthogonality conditions in this approach; cf. [39], [64], [106], and [86]. It is known [39], [63], [64], [41] that a vector equilibrium problem has a unique solution defined by a vector measure $\vec{\lambda}=(\lambda_1,\lambda_2)$ (see (60) and Remark 1 below). Here the zeros of Hermite–Padé polynomials corresponding to the system $[1,f_1,f_2]$ have a limit distribution, which coincides with the measure $\lambda_2$, while for the Hermite–Padé polynomials of the second type corresponding to the pair $f_1$, $f_2$ the limit distribution of zeros coincides with $\lambda_1$. In the alternative approach, proposed originally in [90], an equilibrium problem has to be considered, instead of the Riemann sphere $\widehat{\mathbb{C}}$, on an appropriate two-sheeted Riemann surface. As a result, the equilibrium problem turns out to be scalar, so that a scalar equilibrium measure $\boldsymbol{\lambda}$ occurs, which has now support on the Riemann surface. It was shown in [90] that the limit distribution of the zeros of Hermite–Padé polynomials of the first type coincides with the measure $\lambda=\pi_2(\boldsymbol{\lambda})$, where $\pi_2$ is the canonical projection (two-sheeted covering) of the Riemann surface in question onto the Riemann sphere. As shown in [95], this scalar approach is equivalent to the traditional vector one. It was shown in [45] that the limit distribution of the zeros of Hermite–Padé polynomials of the second type for the same pair of functions $f_1$, $f_2$ forming a Nikishin system as in [90] and [95] can be characterized in terms of the same measure $\boldsymbol{\lambda}$, which has support on a two-sheeted Riemann surface. As already mentioned, in the same example of two Markov functions $f_1$ and $f_2$ as we consider here (see (1) below), it was shown in [95] that these two approaches (the vector and scalar ones) are equivalent. Note that first results on the limit distribution of the zeros of Hermite–Padé polynomials were obtained by Nikishin [63], who used the traditional vector approach, in a much more general situation than here. The idea to use potential theory on a Riemann surface to solve the problem of the limit distribution of zeros of Hermite–Padé polynomials of multivalued analytic functions is not new. The first authors to use potential-theoretic equilibrium problems to construct a suitable three-sheeted Riemann surface were Aptekarev and Kalyagin [8] in 1986. In the general case an approach based on Riemann surfaces was put forward by Stahl in 1987–1988, in his papers [83] and [84]. However, Stahl’s approach has not been developed further, first of all because some of the methods he proposed and some of his results were of heuristic nature, rather than rigorously mathematically substantiated. For more details on Stahl’s approach, see § 5 below, where, in particular, in connection with a result of Stahl (Theorem 4.3 in [84]) we discuss a numerical counterexample (see Example 5). The approach we proposed in [90] is distinct from Stahl’s. In particular, in contrast to our papers [90] and [91], Stahl did not consider in [83] and [84] (cf. [86], [87]) any extremal potential-theoretic problem or equilibrium problem, although he did use potentials on a compact Riemann surface. Nevertheless, for a pair of functions $f_1$, $f_2$, considered both in our original paper [90] and here, these methods turn out to produce the same answer to the question of the so-called weak asymptotic behaviour of Hermite–Padé polynomials of the first and second type alike. We discuss the connection between results obtained by the methods proposed by us [90] (also see [99] and [97]) and by Stahl (‘the third approach’, using a term of Stahl: see [83], § 9) in § 5 below. In this paper we describe the main elements of the two new scalar approaches that we put forward in [90], [99], and [97] to develop them further and apply to extremal and equilibrium problems arising in a natural way in the study of the limit behaviour of the zeros of Hermite–Padé polynomials and the convergence of the corresponding rational approximations. In particular, we show that using the approach proposed in [90] and based on a scalar equilibrium problem on a Riemann surface, we can not just reprove some known results obtained by the classical vector method, but can also obtain new results, which are inaccessible in the framework of the vector approach. Thus, here we discuss two directions of a potential departure from the Gonchar–Rakhmanov vector potential-theoretic problem and their prospects. Namely, we examine this theoretical possibility by considering the example of a pair of functions $(f_1,f_2)$ forming a special (classical) Nikishin system (see (1)) and a pair of functions ($f_1=f$, $f_2=f^2$) forming a generalized Nikishin system, where the algebraic function $f$ is constructed from the inverse Joukowsky function: see (4) and (63). We discuss the possible extensione of these approaches to wider classes of multivalued analytic functions in § 6 below. Note that, as follows from Stahl’s general theory of Padé approximations (see [85], and also [5], [59], and [98]) the problem of the limit distribution of zeros of Hermite–Padé polynomials for a pair of functions consists of two main components, the geometric and analytic ones. Thus, in accordance with Stahl’s approach, in the framework of the first, geometric component, we must prove the existence of a so-called Nuttall condenser (see [75], where this notion was originally introduced). This is an ordered pair of compact sets $\mathrm{N}=(E,F)$ which, for the pair of functions forming a Nikishin system, plays the same role in the asymptotic theory of Hermite–Padé polynomials as the Stahl compact set (admissible compact set of minimum capacity) plays in the asymptotic theory of Padé polynomials. The second, analytic component of Stahl’s approach, in the framework of which the existence of a limit distribution of Padé polynomials is established, is based on the geometric component and the general potential theory in the complex plane. In this paper, in our investigations of the limit distribution of the zeros of Hermite–Padé polynomials for a pair of functions we consider only classes of functions for which the first, geometric component is trivial, namely, the (disjoint) plates the Nuttall condenser lie on the real line and, moreover, consist of a finite number of closed intervals. In the particular potential-theoretic equilibrium problems such that the limit distributions of zeros of Hermite–Padé polynomials of the first and second types are described in terms of their solutions, the $S$-symmetry condition imposed on Nuttal condensers certainly holds in the real case. On the other hand the functions considered in the framework of the two scalar approaches can take complex values on the real line. Thus, the problem of the limit distribution of the zeros of Hermite–Padé polynomials can generally occur outside the range of application of the classical vector approach going back to Gonchar and Rakhmanov [39]. There exist some classes1 of multivalued analytic functions $f$ such that the pair of functions $f$, $f^2$ can naturally be regarded as a complex Nikishin system (see [75], [61], and [91]). In connection with the new approach, presented in [93], to the constructive continuation of an analytic germ (or, in other words — see [14] — an ‘analytic element’, or simply an ‘element’) of a multivalued analytic function we see this fact as one of the main motivations for the examination of extremal potential-theoretic problems and the corresponding equilibrium problems related to general (for instance, complex) Nikishin systems (cf. [15] and [32]). There also exist other applications of Hermite–Padé polynomials to topical problems in various areas of theoretical and applied mathematics; for instance, see [62], [105], [54], and the references there. In particular, much effort has recently been concentrated on Hermite–Padé polynomials for Nikishin systems on star-shaped sets (for instance, see [53]), and on applications of such polynomials to integrable systems [54], [100]. All this clearly shows that a further development of the general theory of Hermite–Padé polynomials, first of all, for Nikishin systems, is a matter of current interest (see [4], [9], [54], [81], [56], [11], and the references there). Overall, many facts on the limit properties of Padé and Hermite–Padé polynomials and the corresponding rational functions are already known. Some of these properties can be illustrated spectacularly in a few important numerical examples, including the analysis of the analytic structure of the frequency functions for the free van der Pol equation; see [2], [28], [1], and [88]. We must point out that the interset to constructive rational approximations (see [44], § 2) increased sharply in the 1970-80s, in connection with the needs of the theory of perturbations in physics and numerical mechanics. Namely, the expansion with respect to a ‘small’ parameter obtained in the framework of perturbation theory must be analysed in some way and then used to find the values of the original function outside the disc of convergence. In other words, the problem of the continuation of a power series outside its disc of convergence was to be solved in some way. Padé approximations turned out to provide a popular method for such purposes; see [17] and also [16], [107], [108], [103], [101], [109] and the bibliography there. We mention in particular that Trias [104] (also see [105]) developed an efficient method called ‘Holomorphic Embedding Load-flow Method’ (HELM-method) to solve the equations of the distribution of load flow in power supply systems. This method is fully based on Padé approximations and Stahl’s theory. Also note that the asymptotic properties of Hermite–Padé polynomials found recently broad applications to Rayleigh–Schrödinger perturbation theory, which holds under the assumptions stated in [48]; for instance, see [30], [31], and the bibliography there. We stress again that all approximations considered in this paper are constructive in the sense of Henrici’s paper [44], § 2 (for instance, cf. [79], [78], [80], [33], [68], [70], and [71]). Namely, they are rational functions not only of the parameter $z$, but also of the initial data of the problem, the first coefficients of the prescribe power series: “… a procedure may be called constructive if it yields the desired mathematical object … as the limit of a single sequence of rational functions of the data of the problem…”. The author is deeply indebted to the referees, who read the manuscript attentively and made some comments leading to the improvement of the presentation of our work. 1.2.In this paper we examine the main properties of two scalar approaches to the investigation of the asymptotic properties of Hermite–Padé polynomials, by taking the example of a pair of functions $f_1$, $f_2$ of a certain form, indicated in § § 1.2.1 and 1.2.2 below. We stress again that in both cases the geometric component of the problem is assumed to be trivial from the outset. Namely, both the plates $E$ and $F$ of the Nuttall condenser $\mathrm N=(E,F)$ (which replaces the Stahl compact set of minimum capacity) lie on the real line. Moreover, $E=[-1,1]$ is just the unit interval and $F=\bigsqcup\limits_{k=1}^q F_k$ consists of a finite number of disjoint closed intervals of the real line. In both cases the pair $(f_1,f_2)$ forms a Nikishin system, classical or generalized, respectively. 1.2.1. Case $1$: a special Nikishin system $f_1$, $f_2$As in [90] (also see [95]), set
$$
\begin{equation}
f_1(z):=\frac{1}{(z^2-1)^{1/2}}\quad\text{and}\quad f_2(z):=\frac{1}{\pi}\int_{-1}^1\frac{h(x)}{z-x}\, \frac{dx}{\sqrt{1-x^2}}\,,\quad z\in D:=\widehat{\mathbb{C}}\setminus E,
\end{equation}
\tag{1}
$$
where $E=[-1,1]$ and where we choose a branch of the root $(\,\cdot\,)^{1/2}$ so that $(z^2-1)^{1/2}/z \to1$ as $z \to\infty$; for $x\in(-1,1)$ we mean by $\sqrt{1-x^2}$ the positive square root: $\sqrt{a^2}=a$ for $a\geqslant0$. Here2 and throughout § § 1 and 2 we assume that the function $h=\widehat{\sigma}$ in (1) is a Markov function, with support $F$ of $\sigma$ equal to a bounded compact subset of the real line disjoint from $E$: $F\subset\mathbb{R}\setminus E$. That is,
$$
\begin{equation}
h(z)=\widehat{\sigma}(z):=\int_{F}\frac{d\sigma(t)}{z-t}\,,\qquad z\in\widehat{\mathbb{C}}\setminus F,
\end{equation}
\tag{2}
$$
where $\sigma$ is a positive Borel measure with support $\operatorname{supp}{\sigma}=F$ such that $\sigma'(t):=d\sigma/dt>0$ almost everywhere on $F$ (see [90], [41], and [38]). We keep this notation throughout the paper. In addition, we assume that $F$ consists of a finite number of disjoint closed intervals: $F=\bigsqcup\limits_{k=1}^q F_k$, $F_k:=[c_k,d_k]$, and the convex hull $\operatorname{conv}(F)$ of $F$ is disjoint from $E$: $\operatorname{conv}(F)\cap E=\varnothing$. Since $f_1$ can be represented in the form
$$
\begin{equation}
f_1(z)=\frac{1}\pi\int_{-1}^1\frac{1}{z-x}\,\frac{dx}{\sqrt{1-x^2}}\,,\qquad z\in D,
\end{equation}
\tag{3}
$$
it follows from (1), (2), and (3) that $\Delta f_2(x)/\Delta f_1(x)=\widehat\sigma(x)$ for $x\in(-1,1)$, where $\Delta f_j(x)=f(x+i0)-f(x-i0)$, $j=1,2$, is the difference of the limit values (jump) of $f_j$ on $(-1,1)$ from the upper and lower half-planes. Hence the pair of functions $(f_1,f_2)$ forms a classical Nikishin system (for more details on such systems, see [63], [64], and also see [9], [18], [55] and the references there). Note that there is a close connection between Hermite–Padé polynomials for a pair of functions forming a Nikishin system and Padé–Chebyshev linear approximations; see [76] and [103]. In their turn, Padé–Chebyshev approximations are widely used in applied problems; for instance, see [20] and [21]. 1.2.2. Case $2$: a pair $f_1=f$, $f_2=f^2$Now assume that a function $f(z)\in{\mathscr H}(\infty)$ is defined explicitly by
$$
\begin{equation}
f(z):=\biggl[\biggl(A_1-\frac{1}{\varphi(z)}\biggr) \biggl(A_2-\frac{1}{\varphi(z)}\biggr)\biggr]^{-1/2},\qquad z\notin E=[-1,1],
\end{equation}
\tag{4}
$$
where $1<A_1<A_2$ and, as before, we choose a branch of the root function $(\,\cdot\,)^{1/2}$ so that $\varphi(z)=z+(z^2-1)^{1/2}\sim 2z$ and $f(z)\sim1/\sqrt{A_1A_2}$ as $z\to\infty$. Then $f$ is an algebraic function of degree four with four first-order branch points $\{\pm1,a_1,a_2\}$, where $a_j=(A_j+1/A_j)/2$, $j=1,2$, $1<a_1<a_2$. The interval $E=[-1,1]$ is a Stahl compact set $S(f)$ for $f$ defined by (4), and $D=\widehat{\mathbb{C}}\setminus{E}$ is the corresponding Stahl domain. We denote the class of such analytic functions by $\mathscr Z_{1/2}(E)$. Note that representation (4) is a special case of the general case (63) discussed in § 3. It was proved in [92] that an arbitrary function $f$ in the class $\in\mathscr Z_{1/2}(E)$ is a Markov function, and both the pair $f$, $f^2$ and the triple $f$, $f^2$, $f^3$ are Nikishin systems (see [63] and [64]):
$$
\begin{equation*}
\begin{alignedat}{2} f(z)&=\frac{1}{\sqrt{A_1A_2}}+\widehat{\sigma}(z),&\qquad \operatorname{supp}{\sigma}&=E, \\ f^2(z)&=\frac{1}{A_1A_2}+\frac{1}{\sqrt{A_1A_2}}\,\widehat{\sigma}(z)+ \widehat{s}_1(z), &\qquad \operatorname{supp}{s_1}&=E,\\ f^3(z)&=\frac{f^2(z)}{\sqrt{A_1A_2}}+\widehat{s}_2(z), &\qquad \operatorname{supp}{s_2}&=E, \end{alignedat}
\end{equation*}
\notag
$$
where $s_1:=\langle \sigma,\sigma_2\rangle$, that is, $ds_1(z)=\widehat{\sigma}_2(z)\,d\sigma(z)$, $\operatorname{supp}{\sigma_2}=[a_1,a_2]$, and $s_2:=\bigl\langle\sigma,\langle\sigma_2,\sigma\rangle\bigr\rangle$. The measures $\sigma$ and $\sigma_2$ have explicit representations; see [92], formulae (16) and (17).
2. First approach2.1.For $n\in\mathbb{N}$ let $\mathbb{P}_n$ denote the space of polynomials of degree $\leqslant n$ with complex coefficients. Given a polynomial $Q\in\mathbb{P}_n^{*}:=\mathbb{P}_n\setminus\{0\}$, we let $\chi(Q)$ denote the counting measure of zeros of $Q$ (taking account of multiplicities): where $\delta_\zeta$ is the unit measure concentrated at the point $\zeta\in{\mathbb{C}}$ (a Dirac delta function).Let $f_1$ and $f_2$ be holomorphic functions at the point at infinity $z=\infty$: $f_j\in{\mathscr H}(\infty)$, $j=1,2$. Throughout what follows $f_1$ and $f_2$ are assumed to have this property. For $n\in\mathbb{N}$ the Hermite–Padé polynomials $Q_{n,0}$, $Q_{n,1}$, and $Q_{n,2}$, $\deg{Q_{n,j}}\leqslant n$, $Q_{n,0}\not\equiv0$, of the first type of degree $n$ for the system of functions $[1,f_1,f_2]$ are defined (not uniquely) in the standard way by the relation
$$
\begin{equation}
R_n(z):=(Q_{n,0}+Q_{n,1}f_1+Q_{n,2}f_2)(z)= O\biggl(\frac{1}{z^{2n+2}}\biggr),\qquad z\to\infty.
\end{equation}
\tag{5}
$$
For $n \in\mathbb{N}$ let $P_{2n,0}$, $P_{2n,1}$, and $P_{2n,2}$, $\deg{P_{2n,j}}\leqslant 2n$, $P_{2n,0}\not\equiv0$, denote the Hermite–Padé polynomials of the second type of degree $2n$ for the pair of functions $(f_1,f_2)$. Namely, these are polynomials defined (not uniquely) by the two relations
$$
\begin{equation}
(P_{2n,0}f_1-P_{2n,1})(z) =O\biggl(\frac{1}{z^{n+1}}\biggr), \qquad z \to\infty,
\end{equation}
\tag{6}
$$
$$
\begin{equation}
\text{and} \qquad (P_{2n,0}f_2-P_{2n,2})(z) =O\biggl(\frac{1}{z^{n+1}}\biggr), \qquad z \to\infty.
\end{equation}
\tag{7}
$$
Now we list some well-known facts on the limit distribution of the zeros of Hermite–Padé polynomials of the first and second type for a pair of functions $f_1$, $f_2$ forming a Nikishin system (see [64], [41], [38], and also [55], [18], and the references in these papers). Let $M_1(E)$ be the class of unit (positive Borel) measures with support on $E$, and let $M_1(F)$ be the analogous class of unit measures with support on $F$. Let $g_F(\zeta,z)$ be the Green’s function for $\Omega:=\widehat{\mathbb{C}}\setminus{F}$ with logarithmic singularity at $\zeta=z$. Let
$$
\begin{equation}
G^\mu_F(z):=\int_E g_F(t,z)\,d\mu(t),\qquad z\in \Omega,
\end{equation}
\tag{8}
$$
be the Green’s potential (with respect to $\Omega$) of the measure $\mu\in M_1(E)$, and let
$$
\begin{equation}
V^\mu(z):=\int_E\log\frac{1}{|z-t|}\,d\mu(t),\qquad z\in D,
\end{equation}
\tag{9}
$$
be the logarithmic potential of $\mu$. It is well known (see [64], Chap. 5; also see [41], [38], [42], and § 7) that there exists a unique measure $\lambda_E\in M_1(E)$ such that $\operatorname{supp}{\lambda_E}=E$ and
$$
\begin{equation}
3V^{\lambda_E}(x)+G^{\lambda_E}_F(x)\equiv c_E=\operatorname{const}
\end{equation}
\tag{10}
$$
on $E$ (identity (10) is the equilibrium relation). Then $\lambda_E$ is called the equilibrium measure for the mixed Green’s-logarithmic potential $3V^\mu(x)+G^\mu_F(x)$, and $c_E$ is the corresponding equilibrium constant. The following result is well known (see [63], [64], [41], [38], and [10]). Theorem 1. Let $f_1$, $f_2$ be a pair of functions defined by (1), where $h(x)=\widehat{\sigma}(x)$, $\operatorname{supp}{\sigma}=F$ and $\sigma'(t)=d\sigma/dt>0$ almost everywhere on $F$. For $n \in \mathbb{N}$ let $P_{2n,0}$ be an Hermite–Padé polynomial of the second type of degree $2n$ defined by (6) and (7). Then $\deg{P_{2n,0}}=2n$ for each $n$, the normalization $P_{2n,0}(z)=z^{2n}+\dotsb{}$ defines $P_{2n,0}$ uniquely, and all zeros of $P_{2n,0}$ are simple and lie in $(-1,1)$. Moreover, In (11) and in what follows we denote by ‘$\xrightarrow{*}$’ weak-$*$ convergence in the space of measures. Note that the above result on the limit distribution of the zeros of Hermite–Padé polynomials of the second type is a very special case of a more general result, proved for a more general setup that the one in Theorem 1 (first of all, see [63], and [64], Chap. 5, § 7, Theorem 7.1; also see [41] and [10]). All these general results were obtained in the framework of the traditional vector approach to the problem of the limit distribution of the zeros of Hermite–Padé polynomials, whose foundations were laid by Gonchar and Rakhmanov in 1981 (see [39]). 2.2.We present the notation and definitions from [90] used in what follows. For $z\in D=\widehat{\mathbb{C}}\setminus E$ let be the inverse Joukowsky functions (recall that throughout we choose a branch of the root function $(\,\cdot\,)^{1/2}$ so that $(z^2-1)^{1/2}/z\to1$ as $z\to\infty$). The function $\varphi$ is single valued and meromorphic in $D$.Let ${\mathfrak R}_2={\mathfrak R}_2(w)$ be the Riemann surface of the function $w^2=z^2-1$. We let the point ${\mathbf z}\in{\mathfrak R}_2$ have the form ${\mathbf z}=(z,w)$. Let $\pi_2\colon{\mathfrak R}_2\to\widehat{\mathbb{C}}$ be a two-sheeted covering of the Riemann sphere $\widehat{\mathbb{C}}$ (the canonical projection): $\pi_2({\mathbf z})=z$. Using the equality $\varphi({\mathbf z})=z+w$ we define $\varphi$ on ${\mathfrak R}_2$. More precisely, set $\Phi({\mathbf z}):=z+w$. Then $\Phi({\mathbf z})$ is the natural analytic continuation of $\varphi(z)$ from the domain $D\subset\widehat{\mathbb{C}}$ to the whole of ${\mathfrak R}_2$. We define a global partition of ${\mathfrak R}_2$ into two open sheets ${\mathfrak R}_2^{(0)}$ (the zeroth sheet) and ${\mathfrak R}_2^{(1)}$ (the first sheet) by setting
$$
\begin{equation*}
z^{(0)}:=\bigl(z,(z^2-1)^{1/2}\bigr)\in{\mathfrak R}_2^{(0)}\quad\text{and}\quad z^{(1)}:=\bigl(z,-(z^2-1)^{1/2}\bigr)\in{\mathfrak R}_2^{(1)}.
\end{equation*}
\notag
$$
As usual, we identify the zeroth sheet ${\mathfrak R}_2^{(0)}$ of ${\mathfrak R}_2$ with the ‘physical’ domain $D=\widehat{\mathbb{C}}\setminus E$ on the Riemann sphere. Set Bearing in mind the above partitioning of ${\mathfrak R}_2$ into sheets we have
$$
\begin{equation*}
\Phi(z^{(0)})=\varphi(z)\quad\text{and}\quad \Phi(z^{(1)})=\frac{1}{\varphi(z)}\,.
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
u(z^{(0)})=-\log|z|+O(1),\quad u(z^{(1)})=\log|z|+O(1),\qquad z\to\infty,
\end{equation*}
\notag
$$
and $u(z^{(0)})<u(z^{(1)})$. Thus the partition under consideration of ${\mathfrak R}_2$ into sheets is the Nuttall partition (see [65], § 3, and [51], Lemma 5). Furthermore,
$$
\begin{equation*}
\pi_2({\mathfrak R}_2^{(0)})=\pi_2({\mathfrak R}_2^{(1)})=D.
\end{equation*}
\notag
$$
Set for ${\mathbf z}\in{\mathfrak R}_2$; in what follows $V({\mathbf z})$ takes the part of an external field3 in the equilibrium problem under consideration. Let $\mathbf F=F^{(1)}\subset{\mathfrak R}_2$ be a compact set on the first sheet ${\mathfrak R}^{(1)}_2$ of ${\mathfrak R}_2$ such that $\pi_2(\mathbf F)=F$ (so that $\mathbf F=F^{(1)}$ is the lift of $F$ to the sheet ${\mathfrak R}_2^{(1)}$ of ${\mathfrak R}_2$).We let $M_1(\mathbf F)$ denote the space of unit (positive Borel) measures with support on $\mathbf F$. Given a measure ${\boldsymbol\mu}\in M_1(\mathbf F)$, following [90] and [95] we introduce a function $P^{\boldsymbol\mu}({\mathbf z})$ of the point ${\mathbf z}\in{\mathfrak R}_2$ (the ‘potential’ of the measure ${\boldsymbol\mu}$; see [95], Remark 2):
$$
\begin{equation}
P^{\boldsymbol\mu}({\mathbf z}):=\int_{\mathbf F}\log \frac{|1-1/(\Phi({\mathbf z})\Phi(\mathbf t))|} {|z-t|^2}\,d{\boldsymbol\mu}(\mathbf t),\qquad {\mathbf z}\in{\mathfrak R}_2\setminus(F^{(0)}\cup F^{(1)}),
\end{equation}
\tag{13}
$$
where $F^{(0)}$ is the lift of $F$ to the zeroth sheet ${\mathfrak R}^{(0)}_2$ of ${\mathfrak R}_2(w)$; we also introduce the corresponding energy of ${\boldsymbol\mu}$ (cf. [25] and [26])
$$
\begin{equation}
J({\boldsymbol\mu}):=\int\!\!\!\int_{\mathbf F\times\mathbf F}\log \frac{|1-1/(\Phi({\mathbf z})\Phi(\mathbf t))|} {|z-t|^2}\,d{\boldsymbol\mu}({\mathbf z})\,d{\boldsymbol\mu}(\mathbf t)= \int_{\mathbf F} P^{\boldsymbol\mu}({\mathbf z})\, d{\boldsymbol\mu}({\mathbf z})
\end{equation}
\tag{14}
$$
relative to the kernel
$$
\begin{equation*}
\log\frac{|1-1/(\Phi({\mathbf z})\Phi(\mathbf t))|}{|z-t|^2}\,.
\end{equation*}
\notag
$$
Also, the energy of ${\boldsymbol\mu}$ in the external field $V({\mathbf z})$ is defined by
$$
\begin{equation}
\begin{aligned} \, J_V({\boldsymbol\mu})&:=\int\!\!\!\int_{\mathbf F\times\mathbf F} \biggl\{\log\frac{|1-1/(\Phi({\mathbf z})\Phi(\mathbf t))|}{|z-t|^2}+ V({\mathbf z})+V(\mathbf t)\biggr\}\,d{\boldsymbol\mu}({\mathbf z})\, d{\boldsymbol\mu}(\mathbf t) \nonumber \\ &\,=\int_{\mathbf F} P^{\boldsymbol\mu}({\mathbf z})\, d{\boldsymbol\mu}({\mathbf z})+2\int_{\mathbf F}V({\mathbf z})\, d{\boldsymbol\mu}({\mathbf z}). \end{aligned}
\end{equation}
\tag{15}
$$
Let $M_1^\circ(\mathbf F)$ denote the set of measures ${\boldsymbol\mu}\in M_1(\mathbf F)$ with finite energy $J_V({\boldsymbol\mu})$.4 Below we identify ${\boldsymbol\mu}\in M_1(\mathbf F)$ and $\mu=\pi_2({\boldsymbol\mu})\in M_1(F)$, where $\pi_2({\boldsymbol\mu})(e):={\boldsymbol\mu}(e^{(1)})$ for each ${\boldsymbol\mu}$-measurable subset $e$ of $ F$. We combine the main results of [90] and [95] into a single statement.5 Proposition 1 (see [90], Theorem 1, and [95], Theorem 1 and Remark 3). There exists a unique scalar measure $\boldsymbol{\lambda}=\boldsymbol{\lambda}_{\mathbf F}\in M_1^\circ(\mathbf F)$ with the following property: In [45] we established the following result by use ot the scalar equilibrium problem (17). Theorem 2. Let $f_1$ and $f_2$ be functions defined by (1), and let $P_{2n,0}$, $n\in\mathbb{N}$, be Hermite–Padé polynomials of the second type defined by (6)–(7). Then $\deg P_{2n,0}=2n$ for all sufficiently large $n$, all zeros of $P_{2n,0}$ lie in $(-1,1)$, and the following limit holds locally uniformly in ${\mathbb{C}}\setminus{E}$: Using Lemma 2 (see § 7 below) and the fact that
$$
\begin{equation*}
\log|\varphi(z)|=g_E(z,\infty)=\gamma_E-V^{\tau_E}(z),
\end{equation*}
\notag
$$
where $g_E(z,\infty)$ is the Green’s function for $D$, $\tau_E$ is the Chebyshev measure for the interval $E$, and $\gamma_E=\log2$ is the Robin constant for $E$, we obtain the following result. Corollary 1. Under the assumptions of Theorem 2 where and $\beta_E(\lambda_F)$ is the balayage of the measure $\lambda_F$ from $D$ to $E=\partial D$. Note that, as follows from § 7, $\lambda_F=\beta_F(\lambda_E)$ is the balayage of $\lambda_E$ from $\Omega=\widehat{\mathbb{C}}\setminus{F}$ to $F=\partial\Omega$. In fact, we obtain representation (20) for $\mu$ by applying the operator $\mathrm{dd}^\mathrm{c}$ to the right-hand side of (18) and using Lemma 2 (see § 7). In accordamce with [95], Theorem 1, the right-hand side of (20) is equal to $\lambda_1=\lambda_E$ (see Theorem 1 above). Thus, relation (19) is equivalent to (11). 2.3.The following result holds. Theorem 3. Let $f_1$ and $f_2$ be functions defined by (1), and let $Q_{n,2}$ be an Hermite–Padé polynomial of the first type defined by (5). Then We stress again that the result of Theorem 3 on the existence of a limit distribution of the zeros of the $Q_{n,2}$ is not new in itself (first of all, see [63], and also [41] and [10]). What is new is the characterization of this limit distribution in terms of the scalar equilibrium problem (16)–(17). We achieve it by setting a suitable potential-theoretic problem on the two-sheeted Riemann surface $w^2=z^2-1$ in place of the Riemann sphere. Proof of Theorem 3. From (5) we obtain
$$
\begin{equation}
0=\int_\gamma(Q_{n,0}+Q_{n,1}f_1+Q_{n,2}f_2)(z)q(z)\,dz= \int_\gamma(Q_{n,1}f_1+Q_{n,2}f_2)(z)q(z)\,dz
\end{equation}
\tag{22}
$$
for each polynomial $q\in\mathbb{P}_{2n}$; $\gamma$ in (22) is an arbitrary contour separating $E$ from the point $z=\infty$.
Let $\operatorname{PA}_{n,0}$ and $\operatorname{PA}_{n,1}$ be the Padé polynomials of degree $n$ for the function $f_1$, so that $\deg{\operatorname{PA}_{n,j}}\leqslant {n}$, $\operatorname{PA}_{n,j}\not\equiv0$, and
$$
\begin{equation}
H_n(z):=(\operatorname{PA}_{n,0}+\operatorname{PA}_{n,1}f_1)(z)= O\biggl(\frac{1}{z^{n+1}}\biggr),\qquad z\to\infty.
\end{equation}
\tag{23}
$$
It is known that in our case the $\operatorname{PA}_{n,1}=T_n$ are the Chebyshev polynomials of the first kind, orthogonal on $E$ with weight $1/\sqrt{1-x^2}$ , and the $H_n$ are the corresponding functions of the second kind. Assume that the Chebyshev polynomials are normalized by $T_n(z)=2^nz^n+\dotsb$. Then for the functions of the second kind we have
$$
\begin{equation}
H_n(z) =\frac{\varkappa_n\varphi'(z)}{\varphi^{n+1}(z)}\,,\quad \varkappa_n\ne0,\quad H_n(z)=\frac{1}{2\pi i}\int_E\frac{T_n(x)\Delta f_1(x)}{x-z}\,dx,\quad z\in D,
\end{equation}
\tag{24}
$$
and
$$
\begin{equation}
\begin{aligned} \, \Delta H_n(x)&:=H_n(x+i0)-H_n(x-i0) \nonumber \\ &\,=T_n(x)\Delta f_1(x)=T_n(x)\frac{2}{i\sqrt{1-x^2}}\,,\qquad x\in(-1,1). \end{aligned}
\end{equation}
\tag{25}
$$
In addition, the polynomials $T_n$ and functions of the second kind $H_n$ satisfy the same second-order recurrence relation, although for different initial data: where we must set $y_{-1}\equiv0$ and $y_0\equiv1$ for the $T_k$, and $y_{-1}\equiv1$, $y_0=f_1(z)=1/(z^2-1)^{1/2}$ for the $H_k$. Since for each polynomial $p\in\mathbb{P}_n$ we have
$$
\begin{equation*}
\int_\gamma p(z)f_1(z)T_{n+j}(z)\,dz=0,\qquad j=1,\dots,n,
\end{equation*}
\notag
$$
from (22), taking $q=T_{n+1},\dots,T_{2n}$ we obtain
$$
\begin{equation}
\int_\gamma Q_{n,2}(z)f_2(z)T_{n+j}(z)\,dz=0,\qquad j=1,\dots,n.
\end{equation}
\tag{27}
$$
Substituting the definition (1) of $f_2$ into (27), changing the order of integration and using Cauchy’s formula we obtain
$$
\begin{equation}
\int_E Q_{n,2}(x)T_{n+j}(x)\frac{1}{\sqrt{1-x^2}}h(x)\,dx=0,\qquad j=1,\dots,n.
\end{equation}
\tag{28}
$$
By (25) this relation is equivalent to
$$
\begin{equation}
\int_\gamma Q_{n,2}(z)H_{n+j}(z)h(z)\,dz=0,\qquad j=1,\dots,n,
\end{equation}
\tag{29}
$$
where $\gamma$ is an arbitrary contour separating $E$ from $F$. As $h(z)=\widehat\sigma(z)$, this relation can easily be reduced to the following form:
$$
\begin{equation}
\int_F Q_{n,2}(x)H_{n+j}(x)\,d\sigma(x)=0,\qquad j=1,\dots,n.
\end{equation}
\tag{30}
$$
These orthogonality relations are key to the further analysis of the existence of a limit distribution of zeros of the polynomials $Q_{n,2}$.
Let $n'$, $0\leqslant n' \leqslant n$, be an arbitrary integer. In what follows we assume without loss of generality that $n'=2m$ is even (the case when $n'$ is odd is quite similar). For arbitrary complex numbers $c_1,\dots,c_{n'}\in{\mathbb{C}}$ consider the sum It is easy to see that since $\deg T_k=k$, using the recurrence relations (26) we can reduce this sum to
$$
\begin{equation}
\sum_{j=1}^{n'}c_jH_{n+j}(z)=q_{m,1}(z)H_{n+m+1}(z)+q_{m,2}(z)H_{n+m}(z),
\end{equation}
\tag{31}
$$
where $q_{m,1},q_{m,2}\in\mathbb{P}_{m-1}$ are polynomial of degree $\leqslant m-1$. Since $c_1,\dots,c_{n'}$ in (31) are arbitrary constants, it is easy to see that we can also select arbitrarily the polynomials $q_{m,1}$ and $q_{m,2}$. In fact, $\deg T_k(z)=k$, so the linear space has dimension $n'$. On the other hand it is contained in of dimension $\leqslant n'$, Therefore, $\operatorname{dim}L_2=n'$ and $L_1=L_2$.
Thus, using (31) we can transform (30) into an equivalent form:
$$
\begin{equation}
\int_F Q_{n,2}(x)\bigl\{q_{m,1}(x)H_{n+m+1}(x)+ q_{m,2}(x)H_{n+m}(x)\bigr\}\,d\sigma(x)=0
\end{equation}
\tag{32}
$$
for arbitrary polynomials $q_{m,1}\in\mathbb{P}_{m-1}$ and $q_{m,2}\in\mathbb{P}_{m-1}$. Now using well-known properties of functions of the second kind (see (24)), from (32) we deduce the following orthogonality relations:
$$
\begin{equation}
\begin{aligned} \, 0&=\int_F Q_{n,2}(x)\biggl\{q_{m,1}(x)\frac{H_{n+m+1}}{H_{n+m}}(x) +q_{m,2}(x)\biggr\}H_{n+m}(x)\,d\sigma(x) \nonumber \\ &=\int_F Q_{n,2}(x)\biggl\{q_{m,1}(x) \frac{\varkappa_{n+m+1}}{\varkappa_{n+m}\varphi(x)}+q_{m,2}(x)\biggr\} \frac{\varkappa_{n+m}\varphi'(x)}{\varphi^{n+m+1}(x)}\,d\sigma(x). \end{aligned}
\end{equation}
\tag{33}
$$
Finally, using the definition of the meromorphic function $\Phi({\mathbf z})$ on the Riemann surface ${\mathfrak R}_2$ (see § 2.2) we obtain the following orthogonality relation for arbitrary polynomials $q_{m,1},q_{m,2}\in\mathbb{P}_{m-1}$: 2.4.Now set
$$
\begin{equation}
g_{n'}({\mathbf z}):=q_{m,1}(z)\Phi({\mathbf z})+q_{m,2}(z),
\end{equation}
\tag{35}
$$
where we assume that $\deg q_{m,1}=\deg q_{m,2}=m-1$. Then for the divisor $\operatorname{div}(g_{n'})$ of $g_{n'}$ we have
$$
\begin{equation}
\operatorname{div}(g_{n'})=-m\infty^{(1)}-(m-1)\infty^{(1)}+ \sum_{j=1}^{{n'}-1}{\mathbf a}_{{n'},j},
\end{equation}
\tag{36}
$$
where it is easy to see that the zeros ${\mathbf a}_{{n'},j}$ of $g_{n'}$ can be arbitrary because $q_{m,1}$ and $q_{m,2}$ are arbitrary polynomials. Since $g_{n'}$ is a meromorphic function on the Riemann surface ${\mathfrak R}_2$ of genus zero, $g_{n'}$ is fully determined by its divisor (of zeros and poles) (36). Hence from (36) we obtain the following explicit representation for $g_{n'}$:
$$
\begin{equation}
g_{n'}({\mathbf z})=C_{n'}\prod_{j=1}^{{n'}-1} \bigl[\Phi({\mathbf z})- \Phi({\mathbf a}_{{n'},j})\bigr]\cdot \Phi({\mathbf z})^{-m+1},\qquad C_{n'}\ne0.
\end{equation}
\tag{37}
$$
In fact, it is easy to verify that the divisor of the right-hand side of (37) coincides with (36). In what follows, in accordance with (34), we are only interested in the case when all points ${\mathbf a}_{n',j}$ lie on the first sheet of ${\mathfrak R}_2$: ${\mathbf a}_{n',j}= a^{(1)}_{n',j}\in{\mathfrak R}^{(1)}$. More precisely, the zeros ${\mathbf a}_{n',j}$ must lie on the first sheet and satisfy $\pi({\mathbf a}_{{n'},j})\in \operatorname{conv}(F)$, where $\operatorname{conv}(F)$ is the convex hull of $F$. In this case, from (37) we obtain
$$
\begin{equation}
g_{n'}(z^{(1)})\Phi(z^{(1)})^{n+m+1}= C_{n'}\prod_{j=1}^{{n'}-1}\bigl[\Phi(z^{(1)})- \Phi(a^{(1)}_{{n'},j})\bigr]\cdot \Phi(z^{(1)})^{n+2}.
\end{equation}
\tag{38}
$$
Now we consider the product $g_{n'}({\mathbf z})\Phi({\mathbf z})^{n+m+1}$ and, using the identity in § 7.3 transform it into the form
$$
\begin{equation}
\begin{aligned} \, g_{n'}({\mathbf z})\Phi({\mathbf z})^{n+m+1}&= C_{n'}\prod_{j=1}^{{n'}-1}\bigl[\Phi({\mathbf z})- \Phi({\mathbf a}_{{n'},j})\bigr]\cdot \Phi({\mathbf z})^{-m+1} \Phi({\mathbf z})^{n+m+1} \nonumber \\ &=\widetilde{C}_{n'}\prod_{j=1}^{{n'}-1} \frac{z-a_{{n'},j}}{1-\Phi({\mathbf z})\Phi({\mathbf a}_{{n'},j})} \cdot\Phi({\mathbf z})^{{n'}+n+1}, \end{aligned}
\end{equation}
\tag{39}
$$
where $\widetilde{C}_{n'}\ne0$ and we assume that all the ${\mathbf a}_{{n'},j}$ are distinct from $\infty^{(0)}$ and $\infty^{(1)}$. In accordance with (34), we need representation (39) only in the case when ${\mathbf z}=z^{(1)}$ and ${\mathbf a}_{{n'},j}=a^{(1)}_{{n'},j}$ for all $j$. In this case, from (39) we obtain
$$
\begin{equation}
g_{n'}(z^{(1)})\Phi(z^{(1)})^{n+m+1}=\widetilde{C}_{n'}\prod_{j=1}^{{n'}-1} \frac{z-a_{{n'},j}}{1-\Phi(z^{(1)})\Phi(a^{(1)}_{{n'},j})} \cdot\Phi(z^{(1)})^{{n'}+n+1}.
\end{equation}
\tag{40}
$$
Since $\Phi(z^{(1)})=1/\varphi(z)$ for all $z\in D$, the last relation can be reduced as follows:
$$
\begin{equation}
g_{n'}(z^{(1)})\Phi(z^{(1)})^{n+m+1}=C_3({n'})\prod_{j=1}^{{n'}-1} \frac{z-a_{{n'},j}}{1-\varphi(z)\varphi(a_{{n'},j})} \cdot\frac{1}{\varphi^{n+2}(z)}\,.
\end{equation}
\tag{41}
$$
Now, taking (41) into account the orthogonality relation (34) assumes the form
$$
\begin{equation}
\int_F Q_{n,2}(x)\prod_{j=1}^{{n'}-1} \frac{x-a_{{n'},j}}{1-\varphi(x)\varphi(a_{{n'},j})} \cdot\frac{\varphi'(x)}{\varphi^{n+2}(x)}\,d\sigma(x)=0,
\end{equation}
\tag{42}
$$
where ${n'}\leqslant {n}$ is arbitrary and all points $a_{{n'},j}$ lie in $D$. It follows from (42) that $\deg{Q_{n,2}}=n$, all zeros of $Q_{n,2}$ lie in the convex hull $\operatorname{conv}(F)$ of the compact set $F=\bigsqcup\limits_{k=1}^q[c_k,d_k]$, and, moreover, the $q-1$ gaps between the intervals $[c_k,d_k]$, $k=1,2,\dots,q$, can contains at most $q-1$ zeros of this polynomial. The orthogonality relations (42), which are defined for an arbitrary ${n'}\leqslant {n}$ and arbitrary points $a_{{n'},j}\in\operatorname{conv}(F)$, underlie the analysis below. 2.5.As is common in the use of the Gonchar–Ralhmanov–Stahl ($\operatorname{GRS}$-)method,7 we will argue by contradiction, that is, we assume that, as $n\to\infty$, Now, from the orthogonality relations (42) and assumption (43) we arrive at a contradiction.As the space of measures $M_1(\operatorname{conv}(F))$ is weakly compact, for some infinite sequence $\Lambda\subset\mathbb{N}$ we have
$$
\begin{equation}
\frac{1}{n}\chi(Q_{n,2})\to\mu\ne\lambda,\qquad n\in\Lambda,\quad n\to\infty,
\end{equation}
\tag{44}
$$
and it follows from the above properties of $Q_{n,2}$ that $\operatorname{supp}{\mu}\subset{F}$, $\mu\in M_1(F)$. We show that (44) and the orthogonality condition (42) are in contradiction. Set
$$
\begin{equation*}
\widetilde{V}^\mu(z):=\int_F\log\frac{1}{|1-\varphi(z)\varphi(t)|}\,d\mu(t);
\end{equation*}
\notag
$$
then Since $\mu\ne\lambda$, for $z\in \operatorname{supp}{\mu}\subset F$ we have
$$
\begin{equation}
P^\mu(z)+\psi(z)\not\equiv m_0:=\min_{z\in F}\bigl(P^\mu(z)+\psi(z)\bigr)= P^\mu(x_0)+\psi(x_0),
\end{equation}
\tag{45}
$$
where Hence there exist $x_1\in \operatorname{supp}{\mu}\subset F$, $x_1\ne x_0$, and $\varepsilon>0$ such that As the function $\psi(z)$ is harmonic in ${\mathbb{C}}\setminus E$ and the potential $P^\mu(z)$ is lower semicontinuous, the same inequality (46) holds in some $\delta$-neighbourhood $U_\delta(x_1):=(x_1-\delta,x_1+\delta)\not\ni x_0$ of $x_1$, $\delta>0$. Since $x_1\in \operatorname{supp}{\mu}$, we have $\mu(U_\delta(x_1))\geqslant\varepsilon_0>0$. Therefore, for all sufficiently large $n\geqslant n_0$, $n\in\Lambda$, there exists a polynomial $p_n(z)=(z-\zeta_{n,1})(z-\zeta_{n,2})$ such that $\zeta_{n,1},\zeta_{n,2}\in U_\delta(x_1)$ and $p_n(z)$ divides $Q_{n,2}$, so that $Q_{n,2}(z)/p_n(z)\in\mathbb{P}_{n-2}$. Set $x_{n-1,n}=\zeta_{n,1}$, $x_{n,n}=\zeta_{n,2}$ and
$$
\begin{equation}
\widetilde{Q}_n(z):=\frac{Q_{n,2}(z)}{p_n(z)}=\prod_{j=1}^{n-2}(z-x_{n,j}).
\end{equation}
\tag{47}
$$
Now we choose ${n'}=n-1$ in (42) and take the zeros $x_{n,j}$ of $\widetilde{Q}_n$ as the $a_{{n'},j}$. Then (42) assumes the following form:
$$
\begin{equation}
\begin{aligned} \, 0&=\int_{F\setminus{U_\delta(x_1)}}\frac{Q_{n,2}^2(x)}{p_n(x)} \prod_{j=1}^{n-2}\frac{1}{1-\varphi(x)\varphi(x_{n,j})}\cdot \frac{\varphi'(x)}{\varphi^{n+2}(x)}\,d\sigma(x) \nonumber \\ &\qquad+\int_{\overline{U}_\delta(x_1)}\frac{Q_{n,2}^2(x)}{p_n(x)} \prod_{j=1}^{n-2}\frac{1}{1-\varphi(x)\varphi(x_{n,j})}\cdot \frac{\varphi'(x)}{\varphi^{n+2}(x)}\,d\sigma(x). \end{aligned}
\end{equation}
\tag{48}
$$
We denote the first integral in (48) by $I_{n,1}$ and the second by $I_{n,2}$. Since $E\cap\operatorname{conv}(F)=\varnothing$, for $x\in F\setminus{U_\delta(x_1)}$ the integrand in $I_{n,1}$ keeps constant sign. Therefore,
$$
\begin{equation}
\begin{aligned} \, |I_{n,1}|&=\int_{F\setminus{U_\delta(x_1)}}\biggl|\frac{Q_{n,2}^2(x)}{p_n(x)} \prod_{j=1}^{n-2}\frac{1}{1-\varphi(x)\varphi(x_{n,j})}\cdot \frac{\varphi'(x)}{\varphi^{n+2}(x)}\biggr|\,d\sigma(x) \nonumber \\ &=\int_{F\setminus{U_\delta(x_1)}}|Q_{n,2}(x)|\prod_{j=1}^{n-2} \biggl|\frac{x-x_{n,j}}{1-\varphi(x)\varphi(x_{n,j})}\biggr|\cdot \frac{\varphi'(x)}{\varphi^{n+2}(x)}\,d\sigma(x). \end{aligned}
\end{equation}
\tag{49}
$$
Using now standard methods of the theory of logarithmic potential, similarly to Lemma 7 in [40] we obtain that there exists a limit
$$
\begin{equation}
\lim_{\substack{n\to\infty\\n\in\Lambda}}|I_{n,1}|^{1/n}= \exp\Bigl\{-\min_{x\in F\setminus{U_\delta(x_1)}} \bigl(P^\mu(x)+\psi(x)\bigr)\Bigr\}=e^{-m_0}.
\end{equation}
\tag{50}
$$
In fact, the proof of (50) is quite analogous to the proof of Lemma 7 in [40]. As $n\to\infty$,
$$
\begin{equation}
-\frac{1}n\sum_{j=1}^{n-2}\log|1-\varphi(x)\varphi(x_{n,j})|\to \int_F\log\frac{1}{|1-\varphi(x)\varphi(t)|}\,d\mu(t)=\widetilde{V}^\mu(x)
\end{equation}
\tag{51}
$$
uniformly in $x\in F$. Hence, as $n\to\infty$,
$$
\begin{equation}
\min_{x\in F}\biggl\{-\frac{1}n\log\biggl(|Q_{n,2}(x)|\prod_{j=1}^{n-2} \biggl|\frac{x-x_{n,j}}{1-\varphi(x)\varphi(x_{n,j})}\biggr|\cdot \biggl|\frac{\varphi'(x)}{\varphi^{n+2}(x)}\biggr|\biggr)\biggr\}\to \min_{x\in F} \bigl\{P^\mu(x)+\psi(x)\bigr\}.
\end{equation}
\tag{52}
$$
This yields the relation
$$
\begin{equation}
\max_{x\in F}\biggl\{|Q_{n,2}(x)|\!\prod_{j=1}^{n-2}\biggl|\frac{x-x_{n,j}} {1-\varphi(x)\varphi(x_{n,j})}\biggr|\cdot \biggl|\frac{\varphi'(x)}{\varphi^{n+2}(x)}\biggr|\biggr\}^{1/n}\!\!\to \exp\Bigl\{-\min_{x\in F}\bigl[P^\mu(x)+\psi(x)\bigr]\Bigr\}
\end{equation}
\tag{53}
$$
as $n\to\infty$, and therefore
$$
\begin{equation*}
\varlimsup_{\substack{n\to\infty\\ n\in\Lambda}} |I_{n,1}|^{1/n} \leqslant e^{-m_0}.
\end{equation*}
\notag
$$
Now we prove the required lower bound. The potential $P^\mu$ is continuous in the fine topology, and therefore $P^\mu+\psi$ is approximately continuous on $F$ with respect to the Lebesgue measure. Hence for each $\varepsilon>0$ the set has a positive Lebesgue measure: $|e|\geqslant\varepsilon_0>0$. Thus it follows that, as $n\to\infty$,
$$
\begin{equation*}
-\frac{1}n\log\biggl\{|Q_{n,2}(x)|\prod_{j=1}^{n-2} \biggl|\frac{x-x_{n,j}}{1-\varphi(x)\varphi(x_{n,j})}\biggr|\cdot \biggl|\frac{\varphi'(x)}{\varphi^{n+2}(x)}\biggr|\biggr\}\to(P^\mu+\psi)(x)
\end{equation*}
\notag
$$
in measure on $F$, and therefore the measure of the set
$$
\begin{equation*}
e_n:=\biggl\{x\in e\colon-\frac{1}{n}\log\biggl(|Q_{n,2}(x)|\prod_{j=1}^{n-2} \biggl|\frac{x-x_{n,j}}{1-\varphi(x)\varphi(x_{n,j})}\biggr|\cdot \biggl|\frac{\varphi'(x)}{\varphi^{n+2}(x)}\biggr|\biggr)< m_0+\varepsilon\biggr\}
\end{equation*}
\notag
$$
tends to the measure of $e$ as $n\to\infty$. Hence
$$
\begin{equation}
\varliminf_{\substack{n\to\infty\\n\in\Lambda}} |I_{n,1}|^{1/n}\geqslant e^{-(m_0+\varepsilon)}\lim_{\substack{n\to\infty\\n\in\Lambda}} \biggl(\int_{e_n}\varphi'(x)\,d\sigma(x)\biggr)^{1/n}=e^{-(m_0+\varepsilon)};
\end{equation}
\tag{54}
$$
the last equality in (54) holds because $\sigma'(x)>0$ almost everywhere on $F$. From (54), since $\varepsilon>0$ can be arbitrary, we obtain the lower bound
$$
\begin{equation*}
\varliminf_{\substack{n\to\infty\\n\in\Lambda}} |I_{n,1}|^{1/n} \geqslant e^{-m_0}.
\end{equation*}
\notag
$$
Relation (50) is proved. On the other hand, for the second integral $I_{n,2}$ we have the upper estimate
$$
\begin{equation}
|I_{n,2}|\leqslant\int_{\overline{U}_\delta(x_1)}|Q_{n,2}(x)|\prod_{j=1}^{n-2} \biggl|\frac{x-x_{n,j}}{1-\varphi(x)\varphi(x_{n,j})}\biggr|\cdot \biggl|\frac{\varphi'(x)}{\varphi^{n+2}(x)}\biggr|\,d\sigma(x).
\end{equation}
\tag{55}
$$
Hence, using arguments similar to the above ones we obtain
$$
\begin{equation}
\begin{aligned} \, \varlimsup_{\substack{n\to\infty\\n\in\Lambda}} |I_{n,2}|^{1/n}\leqslant \exp\Bigl\{-\min_{x\in \overline{U}_\delta(x_1)} \bigl(P^\mu(x)+\psi(x)\bigr)\Bigr\}<e^{-(m_0+\varepsilon)}. \end{aligned}
\end{equation}
\tag{56}
$$
Relations (50) and (56) contradict the equality $I_{n,1}=-I_{n,2}$, which holds by the orthogonality relations (42). $\Box$ 2.6.In the framework of the scalar approach we discuss here, it is possible in principle to investigate the problem in question also in the case when the function $h$ in (1) is complex-valued. Let
$$
\begin{equation}
f_1(z)=\widehat{\sigma}_1(z):=\int_E\frac{d\sigma_1(x)}{z-x}\,,\quad f_2(z):=\int_E\frac{h(x)\,d\sigma_1(x)}{z-x}\,,\qquad z\notin{E},
\end{equation}
\tag{57}
$$
where $\sigma_1$ is a positive measure with support $\operatorname{supp}\sigma_1$ on a compact set $E\subset\mathbb R$, and let $h$ be a holomorphic function on $E$: $h\in{\mathscr H}(E)$. If $h(z)=\widehat{\sigma}_2(z)$, where $\sigma_2$ is a positive measure such that $\operatorname{supp}\sigma_2\subset F$, where $F\subset\mathbb{R}\setminus{E}$ is a compact set, then the pair of functions $f_1$, $f_2$ forms a classical Nikishin system (see [64]; also see [81], [56], [96], [57], and the bibliography there). As before, let $Q_{{n},j}$, $j=0,1,2$, be the Hermite–Padé polynomials of the first type of degree $n$ for the system $[1,f_1,f_2]$. From relation (5) for $h(z)=\widehat{\sigma}_2(z)$ and the functions $f_1$ and $f_2$ defined by (57), we obtain the following orthogonality relation:
$$
\begin{equation}
\int_E(Q_{{n},1}+Q_{{n},2}\widehat{\sigma}_2)(x)x^j\,d\sigma_1(x)=0,\qquad j=0,1,\dots,2n.
\end{equation}
\tag{58}
$$
Let $\operatorname{conv}(E)$ and $\operatorname{conv}(F)$ be convex hulls of the compact sets $E$ and $F$, respectively. If $\operatorname{conv}(E)\cap\operatorname{conv}(F)=\varnothing$, then it follows from (58) that the function
$$
\begin{equation*}
L_{{n}}(z):=Q_{{n},1}(z)+Q_{{n},2}(z)\widehat{\sigma}_2(z)
\end{equation*}
\notag
$$
has at least $2n+1$ zeros $x_{n,k}$, $k=1,\dots,2n+1$, of odd multiplicity on $\operatorname{conv}(E)$. Set $\omega_{2n+1}(z):=\prod\limits_{k=1}^{2n+1}(z-x_{n,k})$. Assume that each of the compact sets $E$ and $F$ consists of a finite number of intervals (and $\operatorname{conv}(E)\cap\operatorname{conv}(F)= \varnothing$). Then it follows from (58) that the polynomials $Q_{{n},2}$ satisfy the orthogonality condition
$$
\begin{equation}
\int_F\frac{Q_{{n},2}(t)t^j}{\omega_{2n+1}(t)}\,d\sigma_2(t)=0,\qquad j=0,1,\dots,n-1.
\end{equation}
\tag{59}
$$
Let $\lambda_1$ and $\lambda_2$ be the unique unit measures with support on $E$ and $F$, respectively, that solve the following vector equilibrium problem (see [39], [64], and [10]):
$$
\begin{equation}
\begin{alignedat}{2} 4V^{\lambda_1}(x)-V^{\lambda_2}(x)&\equiv\operatorname{const},&\qquad x&\in E, \\ V^{\lambda_1}(t)-V^{\lambda_2}(t)&\equiv\operatorname{const},&\qquad t&\in F. \end{alignedat}
\end{equation}
\tag{60}
$$
On the basis of (58) and (59) the following result can be established by using the classical Gonchar–Rakhmanov potential-theoretic method. Theorem 4 (see [39], [64], and [10]). Let $E=\bigsqcup\limits_{j=1}^p E_j$ and $F=\bigsqcup\limits_{k=1}^q F_k$, where $E_j$ and $F_k$ are closed intervals of the real line, be compact sets such that $\operatorname{conv}(E)\cap\operatorname{conv}(F)=\varnothing$. Assume that $\sigma_1'>0$ almost everywhere on $E$ and $\sigma'_2>0$ almost everywhere on $F$. Then, as $n\to\infty$, in the sense of weak-$*$ convergence in the space of measures. Now we modify the geometric component of the problem by abandoning the condition $\operatorname{conv}(E)\cap\operatorname{conv}(F)=\varnothing$ (and replacing it by $E\cap F=\varnothing$). In addition, we extend the class of functions involved in (57) and let $h(z)$ be complex valued. Then we cannot extract from (58) directly that the function $L_{{n}}=Q_{n,1}+Q_{n,2}f$ has zeros. However, for some (quite natural for approximation theory) class of complex functions $h(z)$ relation (58) implies an interesting result on the zeros of $L_{{n}}$. Let $F_k=[c_k,d_k]$, where $c_k<d_k$, $k=1,\dots,q$, let $w^2=\prod\limits_{k=1}^q(z-c_k)(z-d_k)$, and let $w(z)$ be a holomorphic branch of $w$ in ${\mathbb{C}}\setminus{F}$ such that $w(z)/z^q\to1$ as $z\to\infty$. In (57) we set where $r(z)\in{\mathbb{C}}(z)$ is a rational complex-valued function with poles outside $F$, and $r_0(z)$ is the sum of the principal parts of $r(z)/w(z)$ at its poles, so that $h(z)\in{\mathscr H}(\widehat{\mathbb{C}}\setminus{F})$.Then the following result holds (see [96]). Theorem 5. Let $E$ and $F$ be disjoint compact sets consisting of finite numbers of closed intervals of the real line, and let $\sigma_1'>0$ almost everywhere on $E$. In representation (57) for $f_2$ let $h$ be defined by (62). Then there exists a sequence of polynomials $\{\Omega_n\}_{n=n_0}^\infty$ such that (a) for all $n\geqslant n_0$ the functions $(Q_{{n},1}+Q_{{n},2}h)(z)/\Omega_n(z)$ are holomorphic in a fixed neighbourhood of $E$; (b) as $n\to\infty$, in particular, $\deg{\Omega_n}/n\to2$ as $n\to\infty$.The proof of Theorem 5 is based on the $\operatorname{GRS}$-method, which was developed in 1985–87 (see [87], [37], and [40]), and on a new approach to the limit distribution of the zeros of Hermite–Padé polynomials proposed in [91] and founded on potential theory on Riemann surfaces (see [27]). 3. Second approach3.1.As before, let $\varphi(z)=z+(z^2-1)^{1/2}$ be the inverse Joukowsky function, where throughout what follows we choose a branch of the root function $(\,\cdot\,)^{1/2}$ so that $(z^2-1)^{1/2}/z\to1$ as $z\to\infty$ outside $E=[-1,1]$. Hence $\varphi(z)/z\to2$ as $z\to\infty$. Let $m\in\mathbb{N}$, and let $A_j,B_j\in\mathbb{R}$ be real numbers with the following properties:
$$
\begin{equation*}
A_1<B_1<\dots<A_k<B_k<-1\quad\text{and}\quad 1<A_{k+1}<B_{k+1}<\dots<A_m<B_m.
\end{equation*}
\notag
$$
Set
$$
\begin{equation}
w(z):=\prod_{j=1}^m \biggl(\frac{A_j-1/\varphi(z)}{B_j-1/\varphi(z)}\biggr)^{1/2},\qquad z\in D:=\widehat{\mathbb{C}}\setminus{E}.
\end{equation}
\tag{63}
$$
We denote the class of analytic functions with explicit representation (63) by $\mathscr Z(E)$. We stress that the parameters $A_j$ and $B_j$ satisfy by assumption the conditions written above. Note that for $m=1$ the functions $w$, $w^2$ and $w$, $w^2$, $w^3$ form Nikishin systems (see [92]). In accordance with this definition, $w$ is an algebraic function of degree four. All of its branch points have the second order (that is, are quadratic). We denote the set of branch points by $\Sigma$: where
$$
\begin{equation*}
a_j=\frac{1}{2}\biggl(A_j+\frac{1}{A_j}\biggr)\quad\text{and}\quad b_j=\frac{1}{2}\biggl(B_j+\frac{1}{B_j}\biggr),\qquad j=1,\dots,m.
\end{equation*}
\notag
$$
The function $w$ corresponds to a four-sheeted Riemann surface ${\mathfrak R}_4(w)$. Under the above condition on $\varphi(z)$ there exists an analytic element $w_\infty\in{\mathscr H}(\infty)$ of $w$ with the following property:
$$
\begin{equation*}
w_\infty(\infty)=\prod_{j=1}^m\sqrt{\frac{A_j}{B_j}}>0.
\end{equation*}
\notag
$$
This element extends to a holomorphic (single-valued analytic) function in the domain $D=\widehat{\mathbb{C}}\setminus{E}$. Set $F:=\bigsqcup\limits_{j=1}^m[a_j,b_j]$ and $\Omega:=\widehat{\mathbb{C}}\setminus{F}$. Let $f$ be a function in $\mathbb{C}(z,w)$. Then $f $ is a (single-valued) meromorphic function on ${\mathfrak R}_4(w)$. Therefore, the asymptotic properties of the Hermite–Padé polynomials of the first kind for the system of four function $[1,f,f^2,f^3]$ follow directly from [51] (also see [49] and [50]). However, we cannot say this about the triple $[1,f,f^2]$ for $f\in{\mathbb{C}}(z,w)$, since the Riemann surface corresponding to $f$ has four sheets, rather than three. Also note that in general $f\in{\mathbb{C}}(z,w)$ is complex valued on the real line, so in the case under consideration in this section we cannot use the general approach of Gonchar and Rakhmanov. 3.2.Let $f$ be a function in $\mathbb{C}(z,w)$, and let $f_\infty\in{\mathscr H}(\infty)$ be the element of $f$ corresponding to $w_\infty$ selected before. Throughout the end of the section we assume that this condition holds. Fix $n\in\mathbb{N}$, and assume that the polynomials $Q_{n,0},Q_{n,1},Q_{n,2}\in\mathbb{P}_n$, $Q_{n,0}\not\equiv0$, are (non-uniquely) defined by the relation
$$
\begin{equation}
R_n(z):=(Q_{n,0}+Q_{n,1}f_\infty+Q_{n,2}f_\infty^2)(z)= O\biggl(\frac{1}{z^{2n+2}}\biggr),\qquad z\to\infty.
\end{equation}
\tag{64}
$$
Then $Q_{n,0}$, $Q_{n,1}$, and $Q_{n,2}$ are the Hermite–Padé polynomials of the first type of degree $n$ for the system $[1,f_\infty,f_\infty^2]$. The function $R_n(z)=R_n(z;f_\infty)$ is the remainder function. For an arbitrary (positive Borel) measure $\mu$, $\operatorname{supp}{\mu}\subset{\mathbb{C}}$, let $V^\mu(z)$ be, as above, its logarithmic potential, and let
$$
\begin{equation*}
g_F(\zeta,z),\qquad z,\zeta\in\Omega=\widehat{\mathbb{C}}\setminus{F},
\end{equation*}
\notag
$$
be the Green’s function for the domain $\Omega$ with logarithmic singularity at $\zeta=z$, and $G_F^\mu(z)$ be the corresponding Green’s potential (with respect to $\Omega$) of $\mu$. In a similar way we define the Green’s function $g_E(\zeta,z)$ and Green’s potential $G_E^\nu(z)$ for the domain $D=\widehat{\mathbb{C}}\setminus{E}$ and a measure $\nu$ such that $\operatorname{supp}{\nu}\subset{\mathbb{C}}$. It is well known (see [60], [75], and § 7) that there exists a unique probability measure $\lambda_E$ with support on the compact set $E$, $\lambda_E\in M_1(E)$, with the following property:
$$
\begin{equation}
3V^{\lambda_E}(x)+G^{\lambda_E}_F(x)\equiv c_E=\operatorname{const},\qquad x\in E.
\end{equation}
\tag{65}
$$
On the basis of the properties of the potential of the equilibrium measure $\lambda_E$, acting in accordance with the scheme described in [75] (also see [94]), we construct a three-sheeted Riemann surface $\mathscr N_3(f_\infty)$ associated in the sense of Nuttall (relative to the point $\infty^{(0)}$) with the given element $f_\infty$ lifted to ${\mathbf z}=\infty^{(0)}$ (so that $f_{\infty^{(0)}}=f_\infty$). Namely, $\mathscr N_3(f_\infty)$ has the so-called Nuttall partitioning into open sheets $\mathscr N_3^{(0)}\ni z^{(0)}$, $\mathscr N_3^{(1)}\ni z^{(1)}$, and $\mathscr N_3^{(2)}\ni z^{(2)}$ (see [65] and [51]). The given analytic element $f_\infty\in{\mathscr H}(\infty)$ is lifted to the point $\infty^{(0)}$ on the Riemann surface $\mathscr N_3^{(0)}$, after which it extends to the whole Nuttall domain (see [51] and [49]), that is, up to the boundary $F^{(1,2)}$ separating the first sheet $\mathscr N_3^{(1)}$ and the second sheet $\mathscr N_3^{(2)}$ as a (single-valued) meromorphic function. It is easy to see that in our case this element even extends locally across the boundary of the Nuttall domain, to the second sheet $\mathscr N_3^{(2)}$ of $\mathscr N_3(f_\infty)$. The function
$$
\begin{equation}
\begin{alignedat}{2} u(z^{(0)}):&=2V^{\lambda_E}(z)-c_E,&\qquad z&\notin E, \\ u(z^{(1)}):&=-G_F^{\lambda_E}(z)-V^{\lambda_E}(z),&\qquad z&\notin E\sqcup F, \\ u(z^{(2)}):&=G_F^{\lambda_E}(z)-V^{\lambda_E}(z),&\qquad z&\notin{F}, \end{alignedat}
\end{equation}
\tag{66}
$$
extends to the whole of $\mathscr N_3(f_\infty)\setminus\{\infty^{(0)},\infty^{(1)},\infty^{(2)}\}$ as a harmonic function $u({\mathbf z})$ with the following singularities at the points at infinity:
$$
\begin{equation}
\begin{alignedat}{2} u(z^{(0)})&=-2\log|z|+O(1),&\qquad z&\to\infty, \\ u(z^{(1)})&=\log|z|+O(1),&\qquad z&\to\infty, \\ u(z^{(2)})&=\log|z|+O(1),&\qquad z&\to\infty. \end{alignedat}
\end{equation}
\tag{67}
$$
It follows from (66) (see [75] and formulae (74) and (72) in § 4.1 below) that under the assumption that
$$
\begin{equation*}
z^{(0)}\in\mathscr N_3^{(0)}\simeq\widehat{\mathbb{C}}\setminus{E},\quad z^{(1)}\in\mathscr N_3^{(1)}\simeq\widehat{\mathbb{C}}\setminus(E\sqcup F),\quad\text{and}\quad z^{(2)}\in\mathscr N_3^{(2)}\simeq\widehat{\mathbb{C}}\setminus{F}.
\end{equation*}
\notag
$$
Thus, it follows from (67) and (68) that the partition into the three (open) sheets $\mathscr N_3^{(0)}$, $\mathscr N_3^{(1)}$, and $\mathscr N_3^{(2)}$ is indeed the Nuttall partition of $\mathscr N_3(f_\infty)$ relative to the point $\infty^{(0)}$. For details of the construction of $\mathscr N_3(f_\infty)$, see [75], [94], and Fig. 1. Note that in the case of the four-sheeted Riemann surface ${\mathfrak R}_4(w)$ of the function $w$ the structure of its Nuttall partition relative to the point $\infty^{(0)}$ was thoroughly analyzed in [47]. It follows from that paper that under the above conditions on the parameters $A_j$ and $B_j$ the surfaces $\mathscr N_3(w_\infty)$ and ${\mathfrak R}_4(w_\infty)$ have the same zeroth and first sheets of the partitions. However, in general this is not so: for more information, see [47], § 4, and § 6 below. Let $\lambda_F=\beta_F(\lambda_E)\in M_1(F)$ be the balayage of the equilibrium measure $\lambda_E$ from $\Omega=\widehat{\mathbb{C}}\setminus{F}$ to the compact set $F=\partial\Omega$. The following result is central in this section (cf. [93], [51], [49], and [50]). Theorem 6. Let $f\in{\mathbb{C}}(z,w)$ and let $f_\infty\in{\mathscr H}(\infty)$ be selected in accordance with the above conditions on an analytic element $w_\infty$. Then, as $n\to\infty$, the Hermite–Padé polynomials of the first type $Q_{n,j}$, $j=0,1,2$, satisfy and In addition, under the same assumptions, for a certain normalization of the polynomials $Q_{n,j}$ and remainder function (see (105)) the following relations hold as $n\to\infty$:
$$
\begin{equation}
\begin{aligned} \, |Q^{*}_{n,j}(z)|^{1/n}&\xrightarrow{\operatorname{cap}} e^{-V^{\lambda_F}(z)},\qquad z\in\Omega, \nonumber \\ \max_{z\in\Gamma_\rho}\bigl|R_n^{*}(z^{(1)})e^{V^{\lambda_F}(z)}\bigr|^{1/n} &\longrightarrow\frac{1}{\rho^2}\quad\text{for each } \ \rho\in(1,\infty), \end{aligned}
\end{equation}
\tag{71}
$$
where Relation (71) is an analogue of $\rho^2$-results due to Gonchar (see [34]–[36], [82], and [73]). Here ‘$\xrightarrow{\operatorname{cap}}$’ denotes convergence in (logarithmic) capacity on compact subsets of the domain under consideration. The reader can find requisite facts from potential theory on Riemann surfaces in [25]–[27]. 4. Proof of Theorem 64.1.In what follows, for sequences of positive numbers $\{\alpha_n\}$ and $\{\beta_n\}$ the relation $\alpha_n\asymp \beta_n$ means that
$$
\begin{equation*}
0<C_1\leqslant \frac{\alpha_n}{\beta_n}\leqslant C_2<\infty
\end{equation*}
\notag
$$
for all $n=1,2,\dots$ and some constants $C_1$ and $C_2$ independent of $n$. For sequences $\{\alpha_n(z)\}$ and $\{\beta_n(z)\}$ of holomorphic functions in a domain $G$ the relation $\alpha_n\asymp\beta_n$ means that for each compact subset $K$ of $ G$ and all $n=1,2,\dots$ we have
$$
\begin{equation*}
0<C_1\leqslant \biggl|\frac{\alpha_n(z)}{\beta_n(z)}\biggr|\leqslant C_2<\infty
\end{equation*}
\notag
$$
for $z\in K$, where the constants $C_1$ and $C_2$ depend on $K$, but are independent of $n$ and $z\in K$. For such pairs of sequences of numbers or functions we obviously have $|\alpha_n/\beta_n|^{1/n}\to1$ as $n\to\infty$. Let $\mathscr N_3=\mathscr N_3(w_\infty)$ be the three-sheeted Riemann surface associated with a given analytic element $w_\infty$ such that $w\in\mathscr Z(E)$ and $w(\infty)=\prod\limits_{j=1}^m\sqrt{\dfrac{A_j}{B_j}}>0$ in the sense of Nuttall; see [65], [75], [93], and [94]. Recall that the zeroth sheet $\mathscr N_3^{(0)}$ of $\mathscr N_3$ is equivalent to the Riemann surface cut along the line segment $E$: $\mathscr N_3^{(0)}\simeq\widehat{\mathbb{C}}\setminus{E}$. Then $\partial\mathscr N_3^{(0)}=E^{(0,1)}$, where $\pi_3(E^{(0,1)})=E$. The first sheet $\mathscr N_3^{(1)}$ of $\mathscr N_3$ is equivalent to the Riemann sphere cut along the compact sets $E$ and $F=\bigsqcup\limits_{j=1}^m[a_j,b_j]$: $\mathscr N_3^{(1)}\simeq\widehat{\mathbb{C}}\setminus(E\sqcup F)$. We have $\partial\mathscr N_3^{(1)}=E^{(0,1)}\sqcup F^{(1,2)}$, where $\pi_3(F^{(1,2)})=F$. The second sheet $\mathscr N_3^{(2)}$ is equivalent to the Riemann sphere cut along $F$: $\mathscr N_3^{(2)}\simeq\widehat{\mathbb{C}}\setminus{F}$. Then $\partial\mathscr N_3^{(2)}=F^{(1,2)}$, where $\pi_3(F^{(1,2)})=F$; see Fig. 1. Now we prove (68). We have
$$
\begin{equation}
u(z^{(2)})-u(z^{(1)})=2G_F^{\lambda_E}(z)>0,\qquad z\notin F.
\end{equation}
\tag{72}
$$
From [24], formula (16) (also see § 7) we obtain the identity
$$
\begin{equation}
3V^{\lambda_E}(z)+G_F^{\lambda_E}(z)+G_E^{\lambda_F}(z)+3g_E(z,\infty)\equiv c_E=\operatorname{const},\qquad z\in\widehat{\mathbb{C}},
\end{equation}
\tag{73}
$$
where $g_E(z,\infty)$ is the Green’s function for $D$ with logarithmic singularity at the point at infinity. From (73) we obtain
$$
\begin{equation}
\begin{aligned} \, u(z^{(1)})-u(z^{(0)})&=-G_F^{\lambda_E}(z)-3V^{\lambda_E}(z)+c_E \nonumber \\ &=G_E^{\lambda_F}(z)+3g_E(z,\infty)>0,\qquad z\notin{E}. \end{aligned}
\end{equation}
\tag{74}
$$
Relations (68) are direct consequences of (72) and (74). In what follows we need the equality
$$
\begin{equation}
u(z^{(2)})-u(z^{(0)})=2G_F^{\lambda_E}(z)+G_E^{\lambda_F}(z) +3g_E(z,\infty)>0, \qquad z\notin E\sqcup F.
\end{equation}
\tag{75}
$$
4.2.For any $\rho\in(1,\infty)$ let $\Gamma_\rho$ be a level curve of the function $G_F^{\lambda_E}(z)$, namely,
$$
\begin{equation}
\Gamma_\rho:=\{z\in{\mathbb{C}}\colon G_F^{\lambda_E}(z)=\log{\rho}\}.
\end{equation}
\tag{76}
$$
Since $G_F^{\lambda_E}(z)\equiv0$ for $z\in{F}$ and $[a_k,b_k]\cap[a_j,b_j]=\varnothing$ for $k\ne j$, for some $R>1$ and each $\rho\in(1,R]$ the set $\Gamma_\rho$ consists of precisely $m$ disjoint closed curves $(\Gamma_\rho)_j$ such that $\operatorname{int}(\Gamma_\rho)_j\supset F_j$, where $F_j:=[a_j,b_j]$, $j=1,\dots,m$. Set $\Gamma^{(2)}_\rho:=\{z^{(2)}\colon z\in\Gamma_\rho\}$, $\Gamma^{(1)}_\rho:=\{z^{(1)}\colon z\in\Gamma_\rho\}$, and $\Gamma^{(0)}_\rho:=\{z^{(0)}\colon z\in\Gamma_\rho\}$, $\rho\in(1,R]$. We show these sets in Fig. 1 for illustrative purposes. Let $V^{(1,2)}\subset\mathscr N_3$ be a neighbourhood of $F^{(1,2)}$ on $\mathscr N_3$ with the following properties: $\pi_3(\partial V^{(1,2)})=\Gamma_R$, and the analytic element $f_{\infty^{(0)}}$ continues to the domain8
$$
\begin{equation*}
\mathfrak D:=\mathscr N_3^{(0)}\sqcup{E^{(0,1)}}\sqcup \mathscr N_3^{(1)}\cup V^{(1,2)}
\end{equation*}
\notag
$$
as a (single-valued) meromorphic function: $f\in\mathscr M(\mathfrak D)$. It follows immediately from these properties of $V^{(1,2)}\subset\mathscr N_3$ that the remainder function $R_n(z)$ can also be pulled back to $\infty^{(0)}$ and continued to $\mathfrak D$ from this point as a meromorphic function $R_n({\mathbf z})$, ${\mathbf z}\in\mathfrak D$. The function $R_n({\mathbf z})$ has a zero of order at least $2n+2$ at ${\mathbf z}=\infty^{(0)}$, and a pole of order at most $n$ at ${\mathbf z}=\infty^{(1)}$. It can also have poles at some other points in $\mathfrak D$. Let $q_s(z)=z^s+\dotsb$, where $s\in\mathbb{N}$ is fixed, be a polynomial with the following property: $q_sf$ is a holomorphic function in $\mathfrak D\setminus\{\infty^{(0)},\infty^{(1)}\}$. For any $\rho\in(1,R]$ let $\mathfrak D_\rho\subset\mathfrak D$ be the domain in $\mathscr N_3(w_\infty)$ with boundary $\partial\mathfrak D_\rho=\Gamma^{(2)}_\rho$, $\mathfrak D_R=\mathfrak D$. Below, up to § 5 we only look at $\rho\in(1,R]$ such that
$$
\begin{equation*}
q_s(z)f({\mathbf z})\ne0\quad\text{and}\quad f(z^{(0)})-f(z^{(1)})\ne0,\quad\text{for} \ z\in\Gamma_\rho
\end{equation*}
\notag
$$
(it is easy to see that $f(z^{(0)})-f(z^{(1)})\not\equiv0$). We call such values of $\rho$ admissible values. Also assume that this condition also holds for $\rho=R$, so that this value is also admissible. Let $g({\mathbf z}):=-u({\mathbf z})$, ${\mathbf z}\in\mathscr N_3(w_\infty)$. The function $g({\mathbf z})$ is an analogue of the classical $g$-function, although for a three-sheeted Riemann surface (cf. [29], § 7.3, formula (7.46), [47], and [99]). Namely (see (67)),
$$
\begin{equation}
\begin{alignedat}{2} g(z^{(0)})&=c_E-2V^{\lambda_E}(z),&\qquad z&\notin E, \\ g(z^{(1)})&=G_F^{\lambda_E}(z)+V^{\lambda_E}(z),&\qquad z&\notin E\sqcup F, \\ g(z^{(2)})&=V^{\lambda_E}(z)-G_F^{\lambda_E}(z),&\qquad z&\notin{F}. \end{alignedat}
\end{equation}
\tag{77}
$$
Thus,
$$
\begin{equation}
\begin{alignedat}{2} g(z^{(0)})&=2\log|z|+O(1),&\qquad z&\to\infty, \\ g(z^{(1)})&=-\log|z|+O(1),&\qquad z&\to\infty, \\ g(z^{(2)})&=-\log|z|+O(1),&\qquad z&\to\infty, \end{alignedat}
\end{equation}
\tag{78}
$$
and Let $g({\mathbf z};\infty^{(1)},\infty^{(2)})$ be the bipolar Green’s function on $\mathscr N_3$ with singularities at ${\mathbf z}=\infty^{(1)}$ and ${\mathbf z}=\infty^{(2)}$ (see [25]), so that
$$
\begin{equation*}
g({\mathbf z};\infty^{(1)},\infty^{(2)})=\begin{cases} \log|z|+O(1),& {\mathbf z}\to\infty^{(1)}, \\ -\log|z|+O(1),& {\mathbf z}\to\infty^{(2)}. \end{cases}
\end{equation*}
\notag
$$
Let $M\geqslant0$ be the multiplicity of the pole of $f({\mathbf z})$ at ${\mathbf z}=\infty^{(1)}$, and let
$$
\begin{equation*}
r({\mathbf z}):=q_s(z)\exp\{-(2s+M)g({\mathbf z};\infty^{(2)},\infty^{(1)})\};
\end{equation*}
\notag
$$
also let $N=n+1-s$. Set
$$
\begin{equation}
u_n({\mathbf z}):=\log|r({\mathbf z})R_n({\mathbf z})|+Ng({\mathbf z}),\qquad {\mathbf z}\in\mathfrak D.
\end{equation}
\tag{80}
$$
By (78) the function $u_n ({\mathbf z})$ is subharmonic in $\mathfrak D_\rho$ for each $\rho\in(1,R]$ and continuous in a neighbourhood of $\Gamma^{(2)}_\rho$, $\rho\in(1,R]$ (recall that we consider only admissible values of $\rho$). Hence, by the maximum principle for subharmonic functions we have
$$
\begin{equation}
\max_{z\in\Gamma_\rho}u_n(z^{(2)})>\max_{z\in\Gamma_\rho}u_n(z^{(1)}),\qquad \rho\in(1,R].
\end{equation}
\tag{81}
$$
Directly from the definition of $u_n$ (see (80)) we obtain
$$
\begin{equation*}
e^{u_n({\mathbf z})}= \bigl|r({\mathbf z})R_n({\mathbf z})e^{{N}g({\mathbf z})}\bigr|.
\end{equation*}
\notag
$$
Thus, by the definition of the $g$-function and (77) we have
$$
\begin{equation}
\begin{aligned} \, \max_{{\mathbf z}\in\Gamma^{(2)}_\rho}e^{u_n({\mathbf z})}&= \max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R_n(z^{(2)})e^{{N} (V^{\lambda_E}(z)-G_F^{\lambda_E}(z))}\bigr| \nonumber \\ &=\max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R_n(z^{(2)})e^{ {N}V^{\lambda_E}(z)}\bigr|\,\frac{1}{\rho^{N}} \end{aligned}
\end{equation}
\tag{82}
$$
and
$$
\begin{equation}
\begin{aligned} \, \max_{{\mathbf z}\in\Gamma^{(1)}_\rho}e^{u_n({\mathbf z})} &=\max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R_n(z^{(1)}) e^{N(V^{\lambda_E}(z)+G_F^{\lambda_E}(z))}\bigr| \nonumber \\ &=\max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R_n(z^{(1)}) e^{NV^{\lambda_E}(z)}\bigr|\,\rho^{N}. \end{aligned}
\end{equation}
\tag{83}
$$
From (81), (82), and (83) we easily deduce that
$$
\begin{equation}
\max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R_n(z^{(1)}) e^{{N}V^{\lambda_E}(z)}\bigr|<\frac{1}{\rho^{2{N}}} \max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R_n(z^{(2)}) e^{{N}V^{\lambda_E}(z)}\bigr|.
\end{equation}
\tag{84}
$$
In a similar way, applying again the maximum principle to the function $u_n({\mathbf z})$, for $1<\rho<\rho_2\leqslant R$ we obtain
$$
\begin{equation}
\max_{z\in\Gamma_{\rho_2}}\bigl|r({\mathbf z})R_n(z^{(2)}) e^{NV^{\lambda_E}(z)}\bigr|>\biggl(\frac{\rho_2}{\rho}\biggr)^{N} \max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R_n(z^{(2)}) e^{NV^{\lambda_E}(z)}\bigr|.
\end{equation}
\tag{85}
$$
4.3.The following representation is easy to verify:
$$
\begin{equation}
\begin{aligned} \, R_n(z^{(1)})&=R_n(z^{(2)})+\bigl[Q_{n,1}(z)\bigl(f(z^{(1)})-f(z^{(2)})\bigr) +Q_{n,2}(z)\bigl(f^2(z^{(1)})-f^2(z^{(2)})\bigr)\bigr] \nonumber \\ &=R_n(z^{(2)})+\bigl(f(z^{(1)})-f(z^{(2)})\bigr) \bigl[Q_{n,1}(z)+Q_{n,2}(z)\bigl(f(z^{(1)})+f(z^{(2)})\bigr)\bigr], \end{aligned}
\end{equation}
\tag{86}
$$
where $f(z^{(1)})-f(z^{(2)})\ne0$, $z\in\Gamma_\rho$, for each admissible value of $\rho\in(1,R]$. It easily follows from (84) and (86) that for each admissible $\rho$ we have
$$
\begin{equation*}
\begin{aligned} \, &\max_{z\in\Gamma_\rho}\bigl|\bigl[Q_{n,1}(z)+Q_{n,2}(z)\bigl(f(z^{(1)})+ f(z^{(2)})\bigr)\bigr]e^{{N}V^{\lambda_E}(z)}\bigr| \\ &\qquad\asymp\max_{z\in\Gamma_\rho}\bigl|R_n(z^{(2)}) e^{NV^{\lambda_E}(z)}\bigr|,\qquad n\to\infty. \end{aligned}
\end{equation*}
\notag
$$
The maximum principle, as applied to $u_n({\mathbf z})$, easily yields the inequality
$$
\begin{equation}
\max_{z\in\Gamma_\rho}\bigl|R_n(z^{(0)})e^{NV^{\lambda_E}(z)}\bigr|\leqslant Cq^n\max_{z\in\Gamma_\rho}\bigl|R_n(z^{(1)})e^{NV^{\lambda_E}(z)}\bigr|,
\end{equation}
\tag{87}
$$
where $q=q(\rho)<1$. In fact, from the maximum principle for the subharmonic function $u_n({\mathbf z})$ in $\mathfrak D$, for $z\in \Gamma_\rho$ we obtain Using representations (80) for $u_n$ and (77) for $g({\mathbf z})$, from (88) we derive the following relation:
$$
\begin{equation}
\bigl|r({\mathbf z})R_n(z^{(0)})e^{N(c_E-2V^{\lambda_E}(z))}\bigr|< \max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R_n(z^{(1)}) e^{N(G_F^{\lambda_E}(z)+V^{\lambda_E}(z))}\bigr|.
\end{equation}
\tag{89}
$$
Identity (73) yields the following:
$$
\begin{equation}
c_E-2V^{\lambda_E}(z)=V^{\lambda_E}(z)+G_F^{\lambda_E}(z)+ G_E^{\lambda_F}(z)+3g_E(z,\infty).
\end{equation}
\tag{90}
$$
Set
$$
\begin{equation}
q=q(\rho):=\exp\Bigl\{-\min_{z\in \Gamma_\rho}\bigl(G_E^{\lambda_F}(z)+ 3g_E(z,\infty)\bigr)\Bigr\}<1.
\end{equation}
\tag{91}
$$
Directly from (89), (90), and (91) we obtain the required relation (87). From inequality (87) and the identity
$$
\begin{equation}
R_n(z^{(0)})=R_n(z^{(1)})+\bigl(f(z^{(0)})-f(z^{(1)})\bigr) \bigl[Q_{n,1}(z)+Q_{n,2}(z)\bigl(f(z^{(0)})+f(z^{(1)})\bigr)\bigr]
\end{equation}
\tag{92}
$$
(cf. (86)) we see that, as $n\to\infty$,
$$
\begin{equation}
\begin{aligned} \, &\max_{z\in\Gamma_\rho}\bigl|\bigl[Q_{n,1}(z)+Q_{n,2}(z) \bigl(f(z^{(0)})+f(z^{(1)})\bigr)\bigr]e^{{N}V^{\lambda_E}(z)}\bigr| \nonumber \\ &\qquad\asymp \max_{z\in\Gamma_\rho}\bigl|R_n(z^{(1)}) e^{{N}V^{\lambda_E}(z)}\bigr|. \end{aligned}
\end{equation}
\tag{93}
$$
In the same way as above (that is, on the basis of the maximum principle for $u_n({\mathbf z})$) we obtain the following relation:
$$
\begin{equation}
\begin{aligned} \, &\max_{z\in\Gamma_\rho}\bigl|\bigl[Q_{n,1}(z)+Q_{n,2}(z)\bigl(f(z^{(0)})+ f(z^{(2)})\bigr)\bigr]e^{{N}V^{\lambda_E}(z)}\bigr| \nonumber \\ &\qquad\asymp \max_{z\in\Gamma_\rho} \bigl|R_n(z^{(2)})e^{{N}V^{\lambda_E}(z)}\bigr|,\qquad n\to\infty. \end{aligned}
\end{equation}
\tag{94}
$$
From (84), (93), (94), and the identity
$$
\begin{equation*}
\begin{aligned} \, &Q_{n,1}(z)+Q_{n,2}(z)\bigl(f(z^{(0)})+f(z^{(2)})\bigr) \\ &\qquad=Q_{n,1}(z)+Q_{n,2}(z)\bigl(f(z^{(0)})+f(z^{(1)})\bigr)+ Q_{n,2}(z)\bigl(f(z^{(2)})-f(z^{(1)})\bigr), \end{aligned}
\end{equation*}
\notag
$$
since $f(z^{(2)})-f(z^{(1)})\ne0$, for admissible $\rho$ we deduce that
$$
\begin{equation}
\max_{z\in\Gamma_\rho}\bigl|Q_{n,2}(z)e^{{N}V^{\lambda_E}(z)}\bigr|\asymp \max_{z\in\Gamma_\rho}\bigl|R_n(z^{(2)})e^{{N}V^{\lambda_E}(z)}\bigr|.
\end{equation}
\tag{95}
$$
4.4.Since $\lambda_F\in M_1(F)$ is the balayage of $\lambda_E\in M_1(E)$ from the domain $\Omega$ to $F$, we have the identity
$$
\begin{equation}
V^{\lambda_F}(z)=V^{\lambda_E}(z)-G^{\lambda_E}_F(z)+c_0,\qquad z\in\widehat{\mathbb{C}},
\end{equation}
\tag{96}
$$
where the constant satisfies (see [52]). Now note that by (63) the function $q_s(z)(f(z^{(0)})+f(z^{(1)}))$ is holomorphic in the domain $\Omega\setminus\{\infty\}$. This means that
$$
\begin{equation}
v_n(z):=\log\bigl|r({\mathbf z}) \bigl[Q_{n,1}(z)+Q_{n,2}(z)\bigl(f(z^{(0)})+f(z^{(1)})\bigr)\bigr]\bigr|+ NV^{\lambda_F}(z)
\end{equation}
\tag{98}
$$
is a subharmonic function in $\Omega$. Therefore, by the maximum principle and (96), as $\Gamma_\rho$ is a level curve for $G_F^{\lambda_E}(z)$, we obtain the following inequality for $1<\rho<\rho_2\leqslant R$:
$$
\begin{equation*}
\begin{aligned} \, &\max_{z\in\Gamma_{\rho_2}}\bigl|\bigl[Q_{n,1}(z)+Q_{n,2}(z)\bigl(f(z^{(0)})+ f(z^{(1)})\bigr)\bigr]e^{nV^{\lambda_E}(z)}\bigr| \\ &\qquad\leqslant \biggl(\frac{\rho_2}{\rho}\biggr)^n\max_{z\in\Gamma_{\rho}} \bigl|\bigl[Q_{n,1}(z)+Q_{n,2}(z)(f(z^{(0)})+f(z^{(1)}))\bigr] e^{nV^{\lambda_E}(z)}\bigr|. \end{aligned}
\end{equation*}
\notag
$$
By (96) the function
$$
\begin{equation*}
\log|Q_{n,2}(z)|+n\bigl(V^{\lambda_E}(z)-G_F^{\lambda_E}(z)\bigr)
\end{equation*}
\notag
$$
is subharmonic in $\Omega$, and therefore
$$
\begin{equation}
\max_{z\in\Gamma_{\rho_2}}\bigl|Q_{n,2}(z)e^{nV^{\lambda_E}(z)}\bigr| \leqslant \biggl( \frac{\rho_2}{\rho}\biggr)^n \max_{z\in\Gamma_{\rho}}\bigl|Q_{n,2}(z)e^{nV^{\lambda_E}(z)}\bigr|.
\end{equation}
\tag{99}
$$
Note that (99) is an analogue of the Bernstein–Walsh theorem if we consider level curves of the Green’s potential $G_F^{\lambda_E}(z)$ in place of level curves of the Green’s function $g_E(z,\infty)$ (cf. [99]). Finally, combining (85), (95), and (99) for arbitrary admissible $\rho_2,\rho\in(1,R]$ we obtain the following asymptotic relations as $n\to\infty$:
$$
\begin{equation}
\max_{z\in\Gamma_{\rho_2}}\bigl|Q_{n,2}(z)e^{nV^{\lambda_E}(z)}\bigr| \asymp \biggl(\frac{\rho_2}{\rho}\biggr)^n\max_{z\in\Gamma_{\rho}} \bigl|Q_{n,2}(z)e^{nV^{\lambda_E}(z)}\bigr|
\end{equation}
\tag{100}
$$
and
$$
\begin{equation}
\max_{z\in\Gamma_{\rho_2}}\bigl|R_n(z^{(2)})e^{nV^{\lambda_E}(z)}\bigr| \asymp \biggl(\frac{\rho_2}{\rho}\biggr)^n\max_{z\in\Gamma_{\rho}} \bigl|R_n(z^{(2)})e^{nV^{\lambda_E}(z)}\bigr|.
\end{equation}
\tag{101}
$$
4.5.Now we prove that
$$
\begin{equation}
\frac{1}n\chi(Q_{n,2})\xrightarrow{*}\lambda_F,\qquad n\to\infty.
\end{equation}
\tag{102}
$$
For $\rho\in(1,R]$ let
$$
\begin{equation}
m_n(\rho):=\max_{z\in\Gamma_\rho} \bigl|Q_{n,2}(z)e^{nV^{\lambda_E}(z)}\bigr| \quad\text{and}\quad M_{n,2}(\rho):=\max_{z\in\Gamma_\rho} \bigl|R_n(z^{(2)})e^{nV^{\lambda_E}(z)}\bigr|
\end{equation}
\tag{103}
$$
(cf. [99]), where $\Gamma_\rho=\{z\colon G^{\lambda_E}_F(z)=\log\rho\}$. Set
$$
\begin{equation*}
D_\rho:=\{z:G^{\lambda_E}_F(z)<\log\rho\},\quad K_\rho:=\{z:G^{\lambda_E}_F(z)\leqslant \log\rho\},\quad \text{and}\quad \Omega_\rho:=\widehat{\mathbb{C}}\setminus K_\rho.
\end{equation*}
\notag
$$
It follows from (95), (100), and (101) that for $\rho,\rho_2\in(1,R]$ we have
$$
\begin{equation}
m_n(\rho_2)\asymp\biggl(\frac{\rho_2}\rho\biggr)^n m_n(\rho),\ \ M_{n,2}(\rho_2)\asymp\biggl(\frac{\rho_2}\rho\biggr)^n M_{n,2}(\rho),\ \ \text{and}\ \ m_n(\rho)\asymp M_{n,2}(\rho).
\end{equation}
\tag{104}
$$
Now fix $\rho_0\in(1,R]$ and consider the $\rho_0$-normalization (cf. [40], [85], and [25]–[27]) of the polynomials $Q_{n,2}=\prod\limits_{j=1}^{k_n}(z-\zeta_{n,j})$, $k_n=\deg{Q_{n,2}}\leqslant {n}$, relative to the open set $D_{\rho_0}$:
$$
\begin{equation}
Q^{*}_{n,2}(z):=\prod_{\zeta_{n,j}\in D_{\rho_0}}(z-\zeta_{n,j})\cdot \prod_{\zeta_{n,j}\notin D_{\rho_0}}\biggl(1-\frac{z}{\zeta_{n,j}}\biggr).
\end{equation}
\tag{105}
$$
Since relations (104) are invariant under multiplication of both sides by a quantity independent of $z$, they also hold for $Q^{*}_{n,2}$ in place of $Q_{n,2}$. Finally, we can go over from $V^{\lambda_E}$ to the potential $V^{\lambda_F}$ in these relations by using the fact that $\lambda_F$ is the balayage of $\lambda_E$ from the domain $\Omega=\widehat{\mathbb{C}}\setminus{F}$ to $F=\partial\Omega$ (see (96)). As a result, we obtain where $1<\rho<\rho_2\leqslant R$ and
$$
\begin{equation}
m^*_n(\rho):=\max_{z\in\Gamma_\rho} \bigl|Q^*_{n,2}(z)e^{nV^{\lambda_F}(z)}\bigr|.
\end{equation}
\tag{107}
$$
Now set
$$
\begin{equation}
\mu_n:=\frac{1}n\chi(Q_{n,2})= \frac{1}{n}\sum_{j=1}^{k_n}\delta_{\zeta_{n,j}}.
\end{equation}
\tag{108}
$$
Since $\mu_n(\widehat{\mathbb{C}})\leqslant 1$ for all $n$, we can extract a subsequence $\Lambda\subset\mathbb{N}$ such that $\mu_n\xrightarrow{*}\mu$ as $n\to\infty$, $n\in\Lambda$, and $\mu(\widehat{\mathbb{C}})\leqslant 1$. For a measure $\nu$ let $V^*_\nu(z)$ be its $\rho_0$-normalized potential (cf. [40], [85], [25]–[27]):
$$
\begin{equation}
V^*_\nu(z):=\int_{\overline{D}_{\rho_0}}\log\frac{1}{|z-t|}\,d\nu(t)+ \int_{\widehat{\mathbb{C}}\setminus D_{\rho_0}}\log\frac{1}{|1-z/t|}\,d\nu(t).
\end{equation}
\tag{109}
$$
Since the functions $V^*_{\mu_n}$ are superharmonic in ${\mathbb{C}}$ and $V^{\lambda_F}$ is harmonic in $\Omega$, as $n\to\infty$, $n\in\Lambda$, for any $\rho\in(1,R]$ we have
$$
\begin{equation}
\min_{z\in\Gamma_\rho}\bigl\{V^*_{\mu_n}(z)-V^{\lambda_F}(z)\bigr\}\to \min_{z\in\Gamma_\rho}\bigl\{V^*_{\mu}(z)-V^{\lambda_F}(z)\bigr\}.
\end{equation}
\tag{110}
$$
It follows from (106) and (110) that the superharmonic function $v(z):=V^*_\mu(z)-V^{\lambda_F}(z)$ satisfies
$$
\begin{equation}
\min_{z\in\Gamma_\rho} v(z)=\min_{z\in\Gamma_{\rho_2}} v(z)
\end{equation}
\tag{111}
$$
for $1<\rho<\rho_2\leqslant R$. Hence
$$
\begin{equation}
v(z)=V^*_\mu(z)-V^{\lambda_F}(z)\equiv\operatorname{const},\qquad z\in D_R\setminus{F}.
\end{equation}
\tag{112}
$$
Because $\operatorname{supp}{\lambda_F}=F$ and $F$ consists of a finite number of line segments (and therefore has no interior points), we have $\mu=\lambda_F$. Thus we have proved that any limit point of the sequence of measure $\{\mu_n\}$ coincides with $\lambda_F$. In particular, $k_n/n\to1$ as $n\to\infty$, where $k_n=\deg Q_{n,2}$. It follows from what we have proved that for $\rho\in(1,R]$
$$
\begin{equation}
\lim_{n\to\infty}(m^*_n(\rho))^{1/n}= \lim_{n\to\infty}(M^*_{n,2}(\rho))^{1/n}=1,
\end{equation}
\tag{113}
$$
where
$$
\begin{equation}
M^*_{n,2}(\rho)=\max_{z\in\Gamma_\rho} \bigl|R^*_n(z^{(2)})e^{nV^{\lambda_F}(z)}\bigr|.
\end{equation}
\tag{114}
$$
Replacing $V^{\lambda_F}$ by $V^{\lambda_E}$ we set
$$
\begin{equation*}
\widetilde{m}^*_{n}(\rho)=\max_{z\in\Gamma_\rho} \bigl|Q^*_n(z)e^{nV^{\lambda_E}(z)}\bigr|\quad\text{and}\quad \widetilde{M}^*_{n,2}(\rho)=\max_{z\in\Gamma_\rho} \bigl|R^*_n(z^{(2)})e^{nV^{\lambda_E}(z)}\bigr|.
\end{equation*}
\notag
$$
Then bearing in mind that $\lambda_F$ is a balayage of $\lambda_E$, from (114) we obtain
$$
\begin{equation}
\lim_{n\to\infty}(\widetilde{m}^*_n(\rho))^{1/n}= \lim_{n\to\infty}(\widetilde{M}^*_{n,2}(\rho))^{1/n}=\rho e^{-c_0},
\end{equation}
\tag{115}
$$
where From relations (115) and (84), using a suitable analogue of Hadamard’s three-circle theorem on a Riemann surface we obtain
$$
\begin{equation}
\lim_{n\to\infty}(\widetilde{M}^*_{n,1}(\rho))^{1/n}= \frac{1}{\rho}\,e^{-c_0},
\end{equation}
\tag{117}
$$
where
$$
\begin{equation}
\widetilde{M}^*_{n,1}(\rho)=\max_{z\in\Gamma_\rho} \bigl|R^*_n(z^{(1)})e^{nV^{\lambda_E}(z)}\bigr|
\end{equation}
\tag{118}
$$
(cf. (117) and [99], the second relation in (37); to prove that relation one must also use a suitable analogue of Hadamard’s three-circle theorem). 4.6.Now we show that, as $n\to\infty$,
$$
\begin{equation}
\max_{z\in \Gamma_\rho}\bigl|R_n^{*}(z^{(1)})e^{nV^{\lambda_F}(z)}\bigr|^{1/n} \to\frac{1}{\rho^2}\,,\qquad \rho\in(1,R].
\end{equation}
\tag{119}
$$
In fact, let
$$
\begin{equation}
u^{*}_n({\mathbf z}):=\log|r({\mathbf z})R^{*}_n({\mathbf z})|+ Ng({\mathbf z}),\qquad {\mathbf z}\in{\mathfrak D}.
\end{equation}
\tag{120}
$$
Then $u^*_n({\mathbf z})$ is a subharmonic function in ${\mathfrak D}$, which is continuous on $\Gamma^{(j)}_\rho$, $j=0,1,2$, for all admissible $\rho\in(1,R]$. Hence by the maximum principle for subharmonic functions we have
$$
\begin{equation}
\max_{z\in \Gamma_\rho}u^{*}_n(z^{(1)})< \max_{z\in \Gamma_\rho}u^{*}_n(z^{(2)}),\qquad \rho\in(1,R].
\end{equation}
\tag{121}
$$
In accordance with (77), for the function
$$
\begin{equation}
e^{u^{*}_n({\mathbf z})}= \bigl|r({\mathbf z})R^{*}_n({\mathbf z})e^{{N}g({\mathbf z})}\bigr|
\end{equation}
\tag{122}
$$
we obtain
$$
\begin{equation}
\begin{aligned} \, \max_{{\mathbf z}\in \Gamma^{(2)}_\rho}e^{u^{*}_n({\mathbf z})}&= \max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R^{*}_n(z^{(2)}) \exp\bigl\{N\bigl(V^{\lambda_E}(z)-G^{\lambda_E}_F(z)\bigr)\bigr\}\bigr| \nonumber \\ &=\max_{z\in\Gamma_\rho}\bigl|r({\mathbf z})R^{*}_n(z^{(2)}) e^{{N}V^{\lambda_F}(z)}\bigr|e^{-{N}c_0} \end{aligned}
\end{equation}
\tag{123}
$$
because $V^{\lambda_F}(z)=V^{\lambda_E}(z)-G_F^{\lambda_E}(z)+c_0$. Since (see (113))
$$
\begin{equation}
\max_{z\in \Gamma_\rho}\bigl|r({\mathbf z})R^{*}_n(z^{(2)}) e^{NV^{\lambda_F}(z)}\bigr|^{1/n}\to1,\qquad n\to\infty,
\end{equation}
\tag{124}
$$
it follows from (121) and (123) that for all admissible $\rho\in(1,R]$
$$
\begin{equation}
\lim_{n\to\infty}\Bigl(\,\max_{{\mathbf z}\in\Gamma^{(2)}_\rho} e^{u^{*}_n({\mathbf z})}\Bigr)^{1/n}=e^{-c_0}\quad\text{and}\quad \varlimsup_{n\to\infty}\Bigl(\,\max_{{\mathbf z}\in\Gamma^{(1)}_\rho} e^{u^{*}_n({\mathbf z})}\Bigr)^{1/n}\leqslant e^{-c_0}.
\end{equation}
\tag{125}
$$
From (124) and (125), using again a suitable analogue of Hadamerd’s three-circle theorem for Riemann surfaces we see that
$$
\begin{equation}
\lim_{n\to\infty}\Bigl(\,\max_{{\mathbf z}\in\Gamma^{(1)}_\rho} e^{u^{*}_n({\mathbf z})}\Bigr)^{1/n}=e^{-c_0}.
\end{equation}
\tag{126}
$$
By (77), for $z\in\Gamma_\rho$ we have
$$
\begin{equation}
\begin{aligned} \, e^{u^{*}_n(z^{(1)})}&=\bigl|r({\mathbf z})R^{*}_n(z^{(1)}) \exp\bigl\{N\bigl(V^{\lambda_E}(z)+G_F^{\lambda_E}(z)\bigr)\bigr\}\bigr| \nonumber \\ &=\bigl|r({\mathbf z})R^{*}_n(z^{(1)})\exp\bigl\{N\bigl(V^{\lambda_F}(z)+ 2G_F^{\lambda_E}(z)-c_0\bigr)\bigr\}\bigr| \nonumber \\ &=\bigl|r({\mathbf z})R^{*}_n(z^{(1)})e^{{N}V^{\lambda_F}(z)}\bigr|\, \rho^{2{N}}e^{-{N}c_0}. \end{aligned}
\end{equation}
\tag{127}
$$
The required relation (119) follows directly from (126) and (127). 4.7.It is easy to show on the basis of the above that in the interior (that is, on compact subsets) of the domain ${\mathfrak D}$ we have
$$
\begin{equation}
(e^{u^{*}_n({\mathbf z})})^{1/n}\xrightarrow{\operatorname{R-cap}} e^{-c_0},\qquad n\to\infty,
\end{equation}
\tag{128}
$$
where we let ${\operatorname{R-cap}}(\mathbf K)$ denote the Green’s capacity of the compact set $\mathbf K\subset{\mathfrak D}_R$ with respect to $\Gamma^{(2)}_R=\partial{\mathfrak D}_R=\mathfrak D$, $\rho\in(1,R]$ (see [25]–[27]). In fact, as $u^{*}_n({\mathbf z})$ (see (120)) is a subharmonic function in ${\mathfrak D}={\mathfrak D}_R$ which is continuous on $\Gamma^{(2)}_R=\partial{\mathfrak D}_R$, for each compact subset $\mathbf K$ of $ {\mathfrak D}_\rho$, $\rho\in(1,R]$, we have
$$
\begin{equation}
\max_{{\mathbf z}\in \mathbf K}u^{*}_n({\mathbf z})< \max_{{\mathbf z}\in\Gamma^{(2)}_R}u^{*}_n({\mathbf z}).
\end{equation}
\tag{129}
$$
Hence from (125) we obtain
$$
\begin{equation}
\varlimsup_{n\to\infty}\frac{1}{n}\max_{{\mathbf z}\in \mathbf K} u^{*}_n({\mathbf z})\leqslant -c_0,\qquad \mathbf K\subset {\mathfrak D}_\rho.
\end{equation}
\tag{130}
$$
Now we can deduce (128) from (125) and (130) using the two-constant theorem (cf. [40], § 3.8, formulae (31)–(36), [22], and [99]). In fact, (128) means that for any compact set $\mathbf K\subset{\mathfrak D}_\rho$ and each $\varepsilon>0$ the relation
$$
\begin{equation*}
\operatorname{R-cap}\bigl(\mathbf K_{1,n}(\varepsilon)\cup \mathbf K_{2,n}(\varepsilon)\bigr)\to0,\qquad n\to\infty
\end{equation*}
\notag
$$
(cf. [40], § 3.8, formulae (31)–(36)), must hold, where
$$
\begin{equation*}
\begin{aligned} \, \mathbf K_{1,n}(\varepsilon)&:= \bigl\{z\in \mathbf K\colon\bigl(e^{u^{*}_n({\mathbf z})})^{1/n}\geqslant e^{-c_0+2\varepsilon}\bigr\} \\ \text{and} \qquad \mathbf K_{2,n}(\varepsilon)&:= \bigl\{z\in \mathbf K\colon\bigl(e^{u^{*}_n({\mathbf z})})^{1/n}\leqslant e^{-c_0-2\varepsilon}\bigr\}. \end{aligned}
\end{equation*}
\notag
$$
It follows from (130) that we must only consider the case of the set $\mathbf K_n(\varepsilon):=\mathbf K_{2,n}(\varepsilon)$, that is, we have to prove that
$$
\begin{equation*}
\operatorname{R-cap}(\mathbf K_n(\varepsilon))\to0\quad\text{as}\ \ n\to\infty.
\end{equation*}
\notag
$$
Assume the converse: let $\operatorname{R-cap}{\mathbf K_n(\varepsilon)}\geqslant\delta$ for some $\delta>0$ and $n\in\Lambda$, $n\to\infty$. Since $u^{*}_n({\mathbf z})$ is a subharmonic (and therefore upper semicontinuous) function in ${\mathfrak D}_\rho\supset \mathbf K_n(\varepsilon)$, $\rho\in(1,R)$, each point ${\mathbf z}\in \mathbf K_n(\varepsilon)$ has a neighbourhood $U({\mathbf z})$ such that
$$
\begin{equation*}
(e^{u^{*}_n(\boldsymbol\zeta)})^{1/n}< e^{-c_0-\varepsilon},\qquad \boldsymbol\zeta\in U({\mathbf z}),\quad {\mathbf z}\in \mathbf K_n(\varepsilon).
\end{equation*}
\notag
$$
It is easy to see that $\mathbf K_n(\varepsilon)$ is a closed set (see (120)). Hence there exists a compact set $F_n(\varepsilon)=\bigcup\limits_{j=1}^L\overline{U}({\mathbf z}_j)$ such that $F_n(\varepsilon)\supset \mathbf K_n(\varepsilon)$, $F_n(\varepsilon)\subset {\mathfrak D}_\rho$, and, in addition, $F_n(\varepsilon)$ is a regular compact set, $\operatorname{R-cap}(F_n(\varepsilon))\geqslant\delta>0$, $n\in\Lambda$, and
$$
\begin{equation}
\bigl(e^{u^{*}_n({\mathbf z})})^{1/n}\leqslant e^{-c_0-\varepsilon},\qquad {\mathbf z}\in F_n(\varepsilon), \quad n\in\Lambda.
\end{equation}
\tag{131}
$$
Since $u^{*}_n({\mathbf z})$ is a subharmonic function, by the maximum principle we can assume that $F_n(\varepsilon)$ does not separate ${\mathfrak D}_R$. Set
$$
\begin{equation*}
{\mathfrak D}_n(\varepsilon):={\mathfrak D}_R\setminus F_n(\varepsilon).
\end{equation*}
\notag
$$
Then ${\mathfrak D}_n(\varepsilon)$ is a domain with boundary $\partial{\mathfrak D}_n(\varepsilon)= \Gamma^{(2)}_R\cup\partial F_n(\varepsilon)$. Let $\omega_n({\mathbf z})$ be the harmonic measure of the set $\partial F_n(\varepsilon)$ with respect to $\Gamma^{(2)}_R$, that is, $\omega_n({\mathbf z})$ is a harmonic function in ${\mathfrak D}_n(\varepsilon)$ continuous on $\overline{\mathfrak D}_n(\varepsilon)$, and $\omega_n({\mathbf z})\equiv 0$ for ${\mathbf z}\in\Gamma^{(2)}_R$, while $\omega_n({\mathbf z})\equiv1$ for ${\mathbf z}\in\partial F_n(\varepsilon)$. Set
$$
\begin{equation}
w_n({\mathbf z}):=\frac{1}{n}u^{*}_n({\mathbf z})+c_0-\eta- \eta(1-\omega_n({\mathbf z}))+\varepsilon\omega_n({\mathbf z}),
\end{equation}
\tag{132}
$$
where $\eta$ is an arbitrary positive number. Fix some $\rho\in(1,R)$. It follows from (130) and (131) that for $n\in\Lambda$, $n\geqslant n_0(\eta)$, we have
$$
\begin{equation}
w_n({\mathbf z})\leqslant 0,\qquad {\mathbf z}\in {\mathfrak D}_n(\varepsilon).
\end{equation}
\tag{133}
$$
From (132) and (133), for $n\geqslant n_0$ we obtain
$$
\begin{equation}
\bigl(e^{u^{*}_n({\mathbf z})}\bigr)^{1/n}\leqslant \exp\{-c_0-\varepsilon\omega_n({\mathbf z})+\eta+ \eta(1-\omega_n({\mathbf z}))\}
\end{equation}
\tag{134}
$$
uniformly in ${\mathbf z}\in\Gamma^{(2)}_R$. For an arbitrary compact subset $\mathbf K$ of $ {\mathfrak D}_R$ with positive capacity $\operatorname{cap}_{\infty^{(0)}}(\mathbf K)$ with respect to ${\mathbf z}=\infty^{(0)}$ (so that $\mathbf K$ is not polar: see [27], § 5, and cf. [84]) and an arbitrary unit measure ${\boldsymbol\mu}$ with support in $\mathbf K$, $\operatorname{supp}{\boldsymbol\mu}\subset \mathbf K$, we define the Green’s potential of ${\boldsymbol\mu}$ with respect to ${\mathfrak D}_R$ by
$$
\begin{equation}
G^\mu_{{\mathfrak D}_R}({\mathbf z}):=\int g_{{\mathfrak D}_R} (\boldsymbol\zeta,{\mathbf z})\,d\mu(\boldsymbol\zeta).
\end{equation}
\tag{135}
$$
Since $\operatorname{cap}_{\infty^{(0)}}(\mathbf K)>0$, there exists (see [26]) a unique unit measure $\lambda_{\mathbf K}$ with support in $\mathbf K$ such that
$$
\begin{equation}
G^{\lambda_{\mathbf K}}_{{\mathfrak D}_R}({\mathbf z})\equiv \gamma_R(\mathbf K)=\operatorname{const} \quad\text{quasi-everywhere on } \mathbf K.
\end{equation}
\tag{136}
$$
Since $\operatorname{cap}_{\infty^{(0)}}(\mathbf K)>0$, the constant $\gamma_R(\mathbf K)$ is finite, and therefore
$$
\begin{equation}
\operatorname{R-cap}(\mathbf K):=e^{-\gamma_R(\mathbf K)}
\end{equation}
\tag{137}
$$
is a positive quantity. As $F_n(\varepsilon)$ is a regular compact set and $\operatorname{R-cap}(F_n(\varepsilon))\geqslant\delta>0$, it follows that $G^{\lambda_{F_n(\varepsilon)}}_{\mathfrak D_R}({\mathbf z})\equiv \gamma_R(F_n(\varepsilon))$ for ${\mathbf z}\in F_n(\varepsilon)$ and $\gamma_R(F_n(\varepsilon))\leqslant \log(1/\delta)$ for $n\in\Lambda$. Hence the harmonic measure $\omega_n({\mathbf z})$ defined above has the representation
$$
\begin{equation}
\omega_n({\mathbf z})=\frac{1}{\gamma_R(F_n(\varepsilon))} G^{\lambda_{F_n(\varepsilon)}}_{\mathfrak D_R}({\mathbf z}),\qquad {\mathbf z}\in {\mathfrak D}_R(\varepsilon).
\end{equation}
\tag{138}
$$
Let $\rho\in(1,R)$ and
$$
\begin{equation*}
r_n=r_n(\rho):=\min_{{\mathbf z}\in\Gamma^{(2)}_\rho,\, \boldsymbol\zeta\in F_n(\varepsilon)} g_{{\mathfrak D}_R}(\boldsymbol\zeta,{\mathbf z})\geqslant r'>0.
\end{equation*}
\notag
$$
Since $\lambda_{F_n(\varepsilon)}$ is a unit measure, we obtain
$$
\begin{equation}
\min_{{\mathbf z}\in\Gamma_\rho}\omega_n({\mathbf z})\geqslant \frac{r'}{\gamma_R(F_n(\varepsilon))}\geqslant \frac{r'}{\log(1/\delta)}=:r_0>0.
\end{equation}
\tag{139}
$$
It follows from (134) and (139) that
$$
\begin{equation}
\bigl(e^{u^{*}_n({\mathbf z})}\bigr)^{1/n}\leqslant e^{-c_0-\varepsilon r_0+\eta}
\end{equation}
\tag{140}
$$
uniformly in ${\mathbf z}\in \Gamma_\rho$. Hence for $\rho$ such that $q_s(z)\ne0$ on $\Gamma_\rho$ we have
$$
\begin{equation}
\varlimsup_{n\to\infty}\,\max_{{\mathbf z}\in\Gamma^{(2)}_\rho} \bigl(e^{u^{*}_n({\mathbf z})}\bigr)^{1/n}\leqslant e^{-c_0-\varepsilon r_0+\eta},
\end{equation}
\tag{141}
$$
where $\varepsilon>0$ and $r_0>0$ are fixed, while $\eta>0$ can be arbitrary. Now if we let $\eta$ tend to zero in (141), then we arrive at a contradiction with (126). This proves (128). Since $(1/n)\chi(Q_{n,2})\to\lambda_F$, in the interior of $\Omega$ we have
$$
\begin{equation}
|Q^{*}_{n,2}(z)|^{1/n}\xrightarrow{\operatorname{cap}} e^{-V^{\lambda_F}(z)},\qquad n\to\infty.
\end{equation}
\tag{142}
$$
From (93), (128), and (142), in view of the equivalence of convergence in logarithmic and Green’s capacity9 we obtain (70). $\Box$
5. Connection with Stahl’s results5.1.In this subsection we assume that $E=[-1,1]$. Throughout the section, in the definitions of Hermite–Padé polynomials of the first and second type we consider the case when $f_1=f$ and $f_2=f^2$. Moreover, we assume that $f\in\mathscr Z(E)$, where $\mathscr Z(E)$, in conformity with [92] and [94], denotes the class of functions of the form
$$
\begin{equation}
f(z):=\prod_{j=1}^p\biggl(A_j-\frac{1}{\varphi(z)}\biggr)^{\alpha_j},\qquad z\in D=\widehat{\mathbb{C}}\setminus E.
\end{equation}
\tag{143}
$$
Here $p\geqslant2$, the $A_j\in \mathbb{C}$ are pairwise distinct and $|A_j|>1$, $\alpha_j\in \mathbb{C}\setminus{\mathbb Z}$, $j=1,\dots,p$, and $\alpha_1+\cdots+\alpha_p=0$. As before, we take the branch of the root function $(\,\cdot\,)^{1/2}$ such that
$$
\begin{equation*}
\frac{(z^2-1)^{1/2}}{z}\to1\quad\text{and}\quad \frac{\varphi(z)}{z}\to2 \quad\text{as}\ \ z\to\infty.
\end{equation*}
\notag
$$
Functions in $\mathscr Z(E)$ provide a straightforward example of multivalued analytic functions. A function $f\in\mathscr Z({E})$ is multivalued, with branch points
$$
\begin{equation*}
\Sigma=\Sigma(f)=\{\pm 1,a_j,j=1,\dots,p\},\quad\text{where}\ \ a_j:=\frac{1}{2}\biggl(A_j+\frac{1}{A_j}\biggr).
\end{equation*}
\notag
$$
In addition, the branch $f(z)$, $z\in D$, of this function defined by (143) satisfies $f(\infty)=1$ (provided that $\varphi(z)$ behaves as indicated above). We introduce the definitions and notation below by following Stahl [84] (also see [17], § 8.6), but adapt them suitably to those introduced above. Let ${\mathfrak R}$ be a Riemann surface with a finite number of sheets, and let $\pi\colon{\mathfrak R}\to\widehat{\mathbb{C}}$ be the corresponding canonical projection. Thus we regard ${\mathfrak R}$ as a cover of the Riemann sphere with a finite number of sheets (we will also admit an infinite number of sheets in what follows). Hence the set $\pi^{-1}(z)$, $z\in\widehat{\mathbb{C}}\setminus\Sigma$, consists of a finite number of points on ${\mathfrak R}$ lying over the point $z$. Here $\Sigma\subset{\mathbb{C}}$ is the finite set of critical values of $\pi$. Thus, at all points $z\in\widehat{\mathbb{C}}$ (away from the finite set $\Sigma$) the map $\pi$ is locally biholomorphic. As we only consider the pair $f$, $f^2$ and assume that ${\mathfrak R}$ contains sufficiently many (or even an infinite number of) sheets (see representation (143)) Stahl’s condition of ‘covering multiplicity’ introduced in [84] (see condition A, formula (2.3) there) is satisfied. In fact, in our case it means that the following Vandermonde determinant does not vanish identically:
$$
\begin{equation*}
\operatorname{det}(f^j(z^{(k)}))_{j,k=0,1,2}\not\equiv0.
\end{equation*}
\notag
$$
It is easy to see that this is the case in our setting. Hence we can compare Stahl’s heuristic results in [84] with our results in [90], [95], [45], and here, as well as with the results of numerical experiments presented in § 5.5. Throughout the rest of § 5 we assume that $\infty\notin\Sigma$. The given analytic element $f_\infty \in {\mathscr H} (\infty)$, $f_\infty(\infty)=1$, of $f \in \mathscr Z(E)$ extends holomorphically to the domain $D=\widehat{\mathbb{C}}\setminus{E}$, and the interval $E=S$ is the Stahl compact set for $f_\infty \in {\mathscr H}(\infty)$. The two-sheeted Stahl surface $\mathscr S_2(f_\infty)$ corresponding to $f_\infty$ is the Riemann surface ${\mathfrak R}_2={\mathfrak R}_2(w)$ of the function $w^2=z^2-1$. A point ${\mathbf z}$ on ${\mathfrak R}_2(w)$ is a pair $(z,w)$, where $w=\pm(z^2-1)^{1/2}$. This case corresponds to Padé polynomials. It is clear that the Riemann surface of an arbitrary function $f$ in $\mathscr Z({E})$ satisfies all Stahl’s conditions mentioned above; in particular, $\infty\notin\Sigma$ because $\alpha_1+\cdots+\alpha_p= 0$. Taking $\mathscr Z({E})$ as a model class we can compare our results with Stahl’s results and conjectures in [83] and [84]. In particular, we discuss below a few numerical examples of the distribution of the zeros of Hermite–Padé polynomials for functions (143) and functions in the Laguerre class [61]. We let $\infty^{(0)}$ denote the point in the set $\pi^{-1}(\infty)\subset{\mathfrak R}$ at which the function $f$ (defined originally by (143)) takes the value $1$. As is conventional, we identify this point with $\infty$ on the Riemann sphere $\widehat{\mathbb{C}}$, so that we regard $f_\infty$, in fact, as the analytic element $f_{\infty^{(0)}}$, $f_{\infty^{(0)}}(\infty^{(0)})=1$. 5.2.The existence of a limit distribution of the zeros of Padé polynomials $\operatorname{PA}_{n,j}$, $j=1,2$, for any function $f$ in the class $\mathscr Z({E})$ is a consequence of Stahl’s theorem. Moreover, in this quite special case
$$
\begin{equation*}
\frac{1}n\chi(\operatorname{PA}_{n,j})\xrightarrow{*} \tau_{E}(x),\quad n\to\infty,\quad\text{where}\quad d\tau_{E}(x):=\frac{1}{\pi}\,\frac{dx}{\sqrt{1-x^2}}\,.
\end{equation*}
\notag
$$
Following Stahl [84], we set
$$
\begin{equation}
H(z,t):=\begin{cases} z-t, & |t|\leqslant 1, \\ \dfrac{z-t}{|t|}\,, & |t|>1, \\ 1,& t=\infty. \end{cases}
\end{equation}
\tag{144}
$$
For an arbitrary measure $\mu$, $\operatorname{supp}{\mu}\subset\widehat{\mathbb{C}}$, following [84], formula (3.1), we define the spherically normalized potential:10 Let $\mathscr D$ denote the class of domains $\mathfrak D$ on the Riemann surface ${\mathfrak R}={\mathfrak R}(f)$ of $f\in\mathscr Z({E})$, ${\mathfrak D}\subset{\mathfrak R}$, that have the following properties: $\infty^{(0)}\in {\mathfrak D}$, and the domain $\mathfrak D$ can contain at most one point from $\pi^{-1}(\infty)$ other than ${\mathbf z}=\infty^{(0)}$. Also assume that $\infty^{(j)}\notin {\mathfrak D}$ for $j\geqslant1$, and let the Green’s function $g_{\mathfrak D}(\mathbf t,{\mathbf z})$ exist for ${\mathfrak D}$. That is, we assume that the boundary $\partial {\mathfrak D}$ of ${\mathfrak D}$ is non-polar: see [25] and [26]. In what follows we assume that $g_{\mathfrak D}(\mathbf t,{\mathbf z})\equiv 0$ for ${\mathbf z}\in {\mathfrak D}$ and $\mathbf t\in{\mathfrak R}\setminus{\mathfrak D}$. Now let $\nu$ be a signed measure with finite total mass (see [25]) on the Riemann surface ${\mathfrak R}$, and let $\mathfrak D\in\mathscr D$. We define the Green’s potential of $\nu$ (with respect to $\mathfrak D$) in the standard way (see [25] and [26]):
$$
\begin{equation}
G_{\mathfrak D}(\nu;{\mathbf z}):= \int g_{\mathfrak D}(\mathbf t,{\mathbf z})\,d\nu(\mathbf t).
\end{equation}
\tag{146}
$$
For fixed ${\mathbf z}\in{\mathfrak R}$ and a measure $\mu$ on $\widehat{\mathbb{C}}$, following Stahl [84] we set
$$
\begin{equation}
p_\mu({\mathbf z}):=p(\mu;\pi({\mathbf z})),\qquad {\mathbf z}\in{\mathfrak R}.
\end{equation}
\tag{147}
$$
Thus, $p_\mu({\mathbf z})$ is the pullback to ${\mathfrak R}$ of the potential $p(\mu;z)$ of the measure $\mu$ in the complex plane. For an arbitrary integer $m\geqslant1$ the following functions of ${\mathbf z}\in{\mathfrak R}$ were introduced in [84]:
$$
\begin{equation}
\begin{aligned} \, r({\mathbf z})&=r(\mu,\mathfrak D;{\mathbf z}):=G_{\mathfrak D} \bigl((m+1)\delta_{\infty^{(0)}}-\pi^{-1}(\mu);{\mathbf z}\bigr) \nonumber \\ &=(m+1)g_{\mathfrak D}({\mathbf z},\infty^{(0)})- \int g_{\mathfrak D}(\mathbf t,{\mathbf z})\,d\mu(\pi(\mathbf t)) \end{aligned}
\end{equation}
\tag{148}
$$
and It was mentioned in [84] that $m=1$ corresponds to Padé polynomial, so this is the classical case. Hence the first non-trivial case takes place for $m=2$; it corresponds to Hermite–Padé polynomials of the first and second type. Note that in [90] new Hermite–Padé polynomials were introduced for $m=3$ (which are in this case intermediate between the Hermite–Padé polynomials of the first and second type). In [49] and [50] this construction was generalized to all $m\geqslant3$, and relevant results on the convergence of generalized Hermite–Padé polynomials were established. In this paper we consider only $m=2$, that is, we deal with pairs of functions $f_1$, $f_2$. Thus, in contrast to [39], [41], and [10], we discuss only problems stemming from the new approaches in the case of the first non-trivial step (after Padé polynomials corresponding to $m=1$) in the study of the asymptotic properties of Hermite–Padé polynomials. In this case representation (148) for $r({\mathbf z})$ assumes the following form:
$$
\begin{equation}
r({\mathbf z})=3g_{\mathfrak D}({\mathbf z},\infty^{(0)})- \int g_{\mathfrak D}(\mathbf t,{\mathbf z})\,d\mu(\pi(\mathbf t)).
\end{equation}
\tag{150}
$$
The function $r({\mathbf z})$ so defined has the following properties:
The first of Stahl’s results in this direction ([84], Theorem 3.2) reads as follows in the case of our class $\mathscr Z(E)$. Statement 1 (see [84]). Let $f\in\mathscr Z({E})$. Then there exists a unique domain ${\mathfrak D}_1\in \mathscr D$ and a unique unit measure $\nu_1$, $\operatorname{supp}{\nu_1}\subset\widehat{\mathbb{C}}$, such that the function $r({\mathbf z})=r(\nu_1,{\mathfrak D}_1;{\mathbf z})$ extends harmonically from the domain $\mathfrak D_1$ to a neighbourhood of the boundary $\partial {\mathfrak D}_1$ of $\mathfrak D_1$ on the Riemann surface ${\mathfrak R}$. Stahl called the domain ${\mathfrak D}_1\in\mathscr D$ and measure $\nu_1\in M_1(\widehat{\mathbb{C}})$ the convergence domain of type I and the limit distribution of type I, respectively (see [84], Definition 3.3). Stahl also claimed that there exists a limit distribution of the zeros of Hermite–Padé polynomials, which coincides with $\nu_1$ (see [84], Theorem 4.3):
$$
\begin{equation*}
\frac{1}n\chi(Q_{n,j})\xrightarrow{*} \nu_1,\qquad n\to\infty,\quad j=0,1,2.
\end{equation*}
\notag
$$
Thus, the following limit relations holds as $n\to\infty$ in the interior of $D_1:=\pi({\mathfrak D}_1)$:
$$
\begin{equation}
|Q^{*}_{n,j}(z)|^{1/n}\xrightarrow{\operatorname{cap}} e^{p(\nu_1;z)},\qquad z\in\pi({\mathfrak D}_1),\quad j=0,1,2,
\end{equation}
\tag{151}
$$
where the $Q^{*}_{n,j}$ are spherically normalized12 polynomials ($\deg{Q^{*}_{n,j}}/n\to1$), and convergence in (151) is understood as convergence in capacity in the interior (that is, on compact subsets) of $\pi({\mathfrak D}_1)$. Let13
$$
\begin{equation}
d\mu_1({\mathbf z}):=\frac{1}{2\pi}\,\frac{\partial}{\partial n}\, d({\mathbf z})\,ds_{{\mathbf z}},\qquad {\mathbf z}\in\partial{\mathfrak D}_1,
\end{equation}
\tag{152}
$$
where $ds_{{\mathbf z}}$ is the arc length element corresponding to ${\mathbf z}\in\partial\mathfrak D_1$, $\partial/\partial{n}$ is the derivative at ${\mathbf z}\in\partial\mathfrak D_1$ in the inward normal direction to $\partial\mathfrak D_1$, and the function $d({\mathbf z})$ is defined by (149). The following relation establishes a connection between $\mu_1$ and $\nu_1$ (see [84], formulae (3.8) and (3.9b)): Now set
$$
\begin{equation}
\mathfrak D_0:=\{{\mathbf z}\in \mathfrak D_1: r({\mathbf z})> r(\mathbf t)\ \text{for all}\ \mathbf t\in \mathfrak D_1\ \text{such that}\ \pi(\mathbf t)=\pi({\mathbf z}),\ \mathbf t\ne{\mathbf z}\}.
\end{equation}
\tag{153}
$$
Thus, $\mathfrak D_0$ is a subset of $\mathfrak D_1$. Since
$$
\begin{equation*}
r({\mathbf z})=3\log|\pi({\mathbf z})|+O(1)\quad\text{as}\ \ {\mathbf z}\to\infty^{(0)}
\end{equation*}
\notag
$$
and $r({\mathbf z})-\log|\pi({\mathbf z})|$ is a harmonic function in $\mathfrak D_1\setminus\pi^{-1}(\infty)$, it follows that $\infty^{(0)}\in \mathfrak D_0$. From Stahl’s point of view14 the set $\pi(\mathfrak D_0)$ is an open subset of $\widehat{\mathbb{C}}$ such that $\infty\in \pi(\mathfrak D_0)$ and the difference $\widehat{\mathbb{C}}\setminus\pi(\mathfrak D_0)$ consists of a finite number of analytic arcs. Let $\partial/\partial n$ denote the inward normal derivative on $\partial\mathfrak D_0$. Set
$$
\begin{equation}
d\mu_0({\mathbf z}):=\frac{1}{2\pi}\,\frac{\partial}{\partial n} r(\nu_1,\mathfrak D_1;{\mathbf z})\,ds_{{\mathbf z}},\qquad {\mathbf z}\in\partial \mathfrak D_0,
\end{equation}
\tag{154}
$$
where $ds_{{\mathbf z}}$ is the arc length element corresponding to ${\mathbf z}\in \partial \mathfrak D_0$. Since $r({\mathbf z})$ (see (150)) has a third-order logarithmic pole at ${\mathbf z}=\infty^{(0)}$ and is harmonic in $\mathfrak D_0\setminus\{\infty^{(0)}\}$, $\mu_0$ is a positive measure with support on $\partial\mathfrak D_0$, and $\mu_0(\partial\mathfrak D_0)=3$. Set According to Stahl, $\pi(\mathfrak D_0)$ and the measure $\nu_2$ are called a convergence domain of type II and a limit distribution of type II, respectively (see [84], Definition 3.6). From Stahl’s point of view [84], $\mathfrak D_0$ is the ‘zeroth’ sheet15 of the Riemann surface ${\mathfrak R}(f)$, while the open set $\mathfrak D_1\setminus\overline{\mathfrak D}_0$ is its first (open) sheet. Another of Stahl’s results (see [84], Theorem 4.5) is that the limit distribution of the zeros of Harmite–Padé polynomials of the second type exists and coincides with the measure $\nu_2$. Namely, the following result holds. Statement 2 (see [84]). Under the assumptions of Statement 1, the following limit relation holds: 5.3.Now we show that Stahl’s empirical results from [84] cited above and the results in [90], [94], [95], and this paper are perfectly consistent (for the class $\mathscr Z(E)$ under consideration). As in [94], we assume that the quantities $A_j$ and $\alpha_j$, $j=1,\dots,p$, in representation (143) are selected so that for each $j=1,\dots,p$ we have $A_j=\overline{A}_k$ for some $k\in\{1,\dots,p\}$, and the corresponding exponents coincide: $\alpha_j=\alpha_k\in\mathbb{R}\setminus{\mathbb Z}$. If all the $\alpha_j$ belong to $\mathbb{R}\setminus{\mathbb Q}$, then $f$ is an infinite-valued function. Then the conclusions of Theorems 1 and 2 are true (these results were announced in [94]; also see [75]). By [94], Theorem 2, under certain geometric assumptions about the position of the branch points $a_j$ (also see [76]) there exists a compact set $\mathbf F\subset{\mathfrak R}(f)$, $\mathbf F=F^{(1)}$, such that the component of ${\mathfrak R}(f)\setminus\mathbf F$ containing the point ${\mathbf z}=\infty^{(0)}$ is a two-sheeted domain over $ \widehat{\mathbb{C}}$ with boundary equal to a compact set $\mathbf F$ formed by a finite number of analytic arcs. We denote this domain by $\mathfrak{V}_1$, $\partial \mathfrak{V}_1=\mathbf F$. As before (see § 2), let
$$
\begin{equation*}
V({\mathbf z})=\log|z-w|=-\log|\Phi({\mathbf z})|,\qquad {\mathbf z}\in \mathfrak{V}_1,
\end{equation*}
\notag
$$
and
$$
\begin{equation}
P^{\lambda_{\mathbf F}}({\mathbf z})=\int_{\mathbf F}\log \frac{|1-1/(\Phi({\mathbf z})\Phi(\mathbf t))|} {|z-t|^2}\,d\lambda_{\mathbf F}(\mathbf t),\qquad {\mathbf z}\in \mathfrak{V}_1\setminus(F^{(0)}\cup F^{(1)}),
\end{equation}
\tag{157}
$$
where $\lambda_{\mathbf F}\in M_1(\mathbf F)$ is the equilibrium measure for the potential with external field $P^{\lambda_{\mathbf F}}({\mathbf z})+V({\mathbf z})$, that is, $P^{\lambda_{\mathbf F}}({\mathbf z})+V({\mathbf z})\equiv c_{\mathbf F}=\operatorname{const}$ for ${\mathbf z}\in\mathbf F$. Set
$$
\begin{equation*}
u({\mathbf z}):=P^{\lambda_{\mathbf F}}({\mathbf z})+V({\mathbf z})- c_{\mathbf F},\qquad {\mathbf z}\in \mathfrak{V}_1.
\end{equation*}
\notag
$$
The function $u({\mathbf z})$ so defined has the following properties (see [94] and [95]):
$$
\begin{equation*}
u({\mathbf z})=-r(\lambda_{\mathbf F},\mathfrak{V}_1;{\mathbf z})=-r(z).
\end{equation*}
\notag
$$
Hence $\lambda_{\pi(\mathbf F)}=\lambda_F=\nu_1$ and $\mathfrak{V}_1=\mathfrak D_1$. Now consider the set (cf (153))
$$
\begin{equation}
\mathfrak{V}_0:=\{{\mathbf z}\in \mathfrak{V}_1\colon u({\mathbf z})<u(\mathbf t),\mathbf t\in \mathfrak{V}_1,\pi(\mathbf t)= \pi({\mathbf z}),\mathbf t\ne{\mathbf z}\}.
\end{equation}
\tag{158}
$$
Taking [95], § 2.1, into account, it follows from the above properties of $u({\mathbf z})$ that $\mathfrak{V}_0$ is a single-sheeted (with respect to the projection $\pi$) subdomain of $\mathfrak{V}_1$ containing ${\mathbf z}=\infty^{(0)}$, which has the boundary $\pi^{-1}({E})\cap\mathfrak {V}_1$. In other words,
$$
\begin{equation*}
\mathfrak{V}_0=\{{\mathbf z}=(z,(z^2-1)^{1/2}),z\in D= \widehat{\mathbb{C}}\setminus{E}\}.
\end{equation*}
\notag
$$
Since $u({\mathbf z})=-r({\mathbf z})$, it follows that $\mathfrak{V}_0=\mathfrak D_0$. Now it is easy to see that Stahl’s measure $\nu_2$ from [84] coincides with $\lambda_E$ (see (10)). Thus, the results in [90], [94], [95], and [45] are perfectly consistent with Stahl’s ones in [84]. On the other hand note that Stahl’s results in [83] and [84] are largely based on heuristic ideas, rather than on rigorous mathematical arguments. For this reason, some of Stahl’s results in those papers should better be viewed as conjectures, rather than rigorously established assertions. In § 5.5 below we present and discuss a numerical example related to a system $[1,f,f^2]$ for some algebraic function of degree five and to the corresponding Hermite–Padé polynomials of the first type, which contradicts one of Stahl’s statements (Theorem 3.4 in [84]). Note that Nuttall’s well-known paper [65] of 1984 contains both rigorously proved results and results established heuristically, which were subsequently regarded as conjectures requiring proofs: see [50] and [51]. 5.4.In conclusion, we discuss the possible direction of further development of the asymptotic theory of Hermite–Padé polynomials for a pair of multivalued analytic functions $f_1$, $f_2$, and of rational approximations corresponding to such polynomials in the case when $f_1=f$ and $f_2=f^2$. We stress again that Stahl’s results in [83] and [84] are mostly heuristic and must be substantiated rigorously, also in the case when $f_1=f$ and $f_2=f^2$. The advantage of the first scalar approach over the classical vector one can be explained as follows. As with Padé polynomials, in the scalar approach we prove the existence of a unique extremal compact set, which now lies on a two-sheeted Riemann surface. This is also clear from the numerical results, presented below, on the identification of zeros of Hermite–Padé polynomials of the first and second type for functions of the form (143) and some more general functions. Of course, in Examples 1–3 below all the $\alpha_j$ in (143) are rational numbers. We note a fact related to Examples 1 and 2. In both examples the parameter $p$ in (143) is equal to 2, that is, apart from $z=\pm1$ the functions represented by (143) have only the branch points $a_1=(A_1+1/A_1)/2$ and $a_2=(A_2+1/A_2)/2$. However, if we consider these functions as defined at the point ${\mathbf z}=\infty^{(0)}$ on the zeroth sheet of the two-sheeted Riemann surface $w^2=z^2-1$, then they turn out to have only two branch points, $a^{(1)}_1$ and и $a^{(1)}_2$, $a^{(1)}_j\in{\mathfrak R}_2^{(1)}$, $\pi_2(a^{(1)}_j)=a_j$, $j=1,2$. Since both of these branch points are of the second order, the complement to any arc joining $a^{(1)}_1$ to $a^{(1)}_2$ is a domain of single-valued analytic continuation of the element $f_{\infty^{(0)}}$. Taking such a class of ‘admissible’ compact sets we occur naturally in the situation when the extremal compact set $\mathbf F=F^{(1)}\subset{\mathfrak R}_2^{(1)}(w)$ (see § 5.3) is not the line segment $[a_1^{(1)},a_2^{(2)}]$ any longer, but is an analytic arc joining $a^{(1)}_1$ and $a^{(1)}_2$. When we consider Padé approximations for a function with a pair of second-order branch points, the arc in question is always the line segment connecting these branch points, for whatever positions of these points relative to the point at infinity $z=\infty$ at which the initial element is prescribed. This is because, in accordance with the statement of the problem for Padé approximations, the extremal compact set is the set of minimum capacity (relative to $z=\infty$). On the other hand, in the case of Hermite–Padé polynomials we have another extremal problem (relative to $\infty^{(0)}$; cf. [47], where the Nuttall partition of the four-sheeted Riemann surface of a function of the form (4) was investigated). Namely, in the case when the branch points $a_1$ and $a_2$ in (4) do not lie on the real line (see Examples 1 and 2 below; also see Examples 3 and 4) the extremal compact set $\mathbf F$ does not any longer lie on the real line either, is not known in advance and, as in the classical case of Padé approximations and the compact set of minimum capacity, is defined as a solution of a ‘$\max$-$\min$’-problem in a suitable class of admissible compact sets (see [69], [91], Theorem 2, and [94], Theorem 2). The logarithmic potential and energy must now be replaced by the energy (15) of the potential (13) with an external field. The extremal compact set $\mathbf F=F^{(1)}\subset{\mathfrak R}_2(w)$ gives rise to the compact set $F:=\pi_2(\mathbf F)\subset\widehat{\mathbb{C}}$, which is the second plate of the Nuttall condenser. The first plate $E$ of this condenser is now uniquely defined by $F$. Namely (see [45], formula (2.43)), $E=\pi_2({\mathbf E})$, where
$$
\begin{equation}
{\mathbf E}=\biggl\{{\mathbf z}\in{\mathfrak R}_2(w)\colon \int_{\mathbf F}\log\biggl|\frac{1-\Phi({\mathbf z})\Phi({\boldsymbol\zeta})} {\Phi({\mathbf z})-\Phi({\boldsymbol\zeta})}\biggr|\, d\boldsymbol{\lambda}_{\mathbf F}({\boldsymbol\zeta})+3V({\mathbf z})= 0\biggr\};
\end{equation}
\tag{159}
$$
here $\boldsymbol{\lambda}_{\mathbf F}$ is the equilibrium measure for $\mathbf F$ (see Corollary 1 and relation (194) after it for $\theta=3$). Note that, in contrast to $\mathbf F=F^{(1)}\subset{\mathfrak R}_2(w)$, which is an analytic arc connecting $a_1^{(1)}$ with $a_2^{(1)}$, the compact set ${\mathbf E}\subset{\mathfrak R}_2(w)$ is a closed analytic arc passing through the points $\pm1$. As mentioned in [94], from $\mathbf F\subset{\mathfrak R}_2(w)$ and the potential (13) we can construct a three-sheeted Riemann surface with Nuttall partition that is associated with the given analytic element $f_{\infty^{(0)}}$, and the compact sets ${\mathbf E}$ and $\mathbf F$ correspond to boundaries between Nuttall sheets. 5.5. Some examples Example 1. In this example, in representation (143) we have  Figure 2.Black dots show zeros of the Hermite–Padé polynomial of the first type $Q_{100,2}$ for the analytic element $f_\infty$ of the function with representation (143) for $p=2$, $A_2=\overline{A}_1 \notin \mathbb{R}$, and $\alpha_1=\alpha_2=-1/2$. They model the second plate $F$ of the Nuttall condenser. Light blue dots show zeros of the Hermite–Padé polynomial of the second type $P_{200,0}$, which model the first plate $E$ of the Nuttall condenser.  Figure 3.Black dots show zeros of the Hermite–Padé polynomial of the first type $Q_{100,2}$ for the function with representation (143) for $p=2$, $A_1 \ne A_2$, $\operatorname{Im} A_1,\operatorname{Im} A_2>0$, and $\alpha_1=\alpha_2=-1/2$. They model the second plate $F$ of the Nuttall condenser. Light blue dots are zeros of the Hermite–Padé polynomial of the second type $P_{200,0}$, which model the first plate $E$ of the Nuttall condenser. In this case the zeros of the Hermite–Padé polynomials of the first type $Q_{n,0}$ and $Q_{n,1}$ have the same distribution as those of $Q_{n,2}$ (cf. Example 5).  Figure 4.Black dots show zeros of the Hermite–Padé polynomial of the first type $Q_{100,2}$ for the function with representation (143) for $p=3$, $A_1 \in \mathbb{R}$, $A_2=\overline{A}_3 \notin \mathbb{R}$, $\alpha_1=-2/3$, and $\alpha_2=\alpha_3=1/3$. They model the second plate $F$ of the Nuttall condenser. Light blue dots are zeros of the Hermite–Padé polynomial of the second type $P_{200,0}$, which model the first plate $E$ of the Nuttall condenser. In this case the zeros of the Hermite–Padé polynomials of the first type $Q_{n,0}$ and $Q_{n,1}$ have the same distribution as those of $Q_{n,2}$ (cf. Example 5).  Figure 5.Blue, red, and black dots correspond to zeros of the Hermite–Padé polynomials of the first type $Q_{350,0}$, $Q_{350,1}$, and $Q_{350,2}$, respectively, for the system $[1,f_\infty,f_\infty^2]$, where $f_\infty$ is the analytic element of the function of the form (160) for $p=2$, $A_1=\overline{A}_2 \notin \mathbb{R}$ and $q=3$, $B_1=\overline{B}_2 \notin \mathbb{R}$, $B_3 \in \mathbb{R}$. These zeros model the compact set $F=\pi_2(\mathbf{F})$. Light blue points are zeros of the Hermite–Padé polynomial of the second type $P_{200,0}$, which model the compact set $E=\pi_2(\mathbf{E})=E_1 \sqcup E_2$. The pair $(E,F)$ forms the Nuttall condenser. The number of Chebotarev points with density zero is 5; it is equal to the number of branch points distinct from $e_1$, $e_2$, $e_3$, and $e_4$, which are the endpoints of the intervals $E_1$ and $E_2$. Example 2. In this example, in representation (143) we have Example 3. In this example, in representation (143) we set Example 4. In this example we consider a more general class of multivalued analytic functions than $\mathscr Z({E})$. Namely, in place of one interval $E=[-1,1]$ we consider two disjoint intervals of the real line $E_1$ and $E_2$, and the two inverse Joukowsky functions $\varphi_{{E}_1}(z)$ and $\varphi_{{E}_2}(z)$ corresponding to them. In this case the multivalued function $f$ has the form Let $e_1$, $e_2$ and $e_3$, $e_4$, $e_1<e_2<e_3<e_4$, be the endpoints of $E_1$ and $E_2$, respectively, and ${\mathfrak R}_2(w)$ be the two-sheeted elliptic Riemann surface of the function $w^2=(z-e_1)(z-e_2)(z-e_3)(z-e_4)$; let $\pi_2\colon{\mathfrak R}_2(w)\to\widehat{\mathbb{C}}$ be the corresponding canonical projection. Under our assumptions on the parameters of the function in (160), in accordance with the ‘general’ heuristic arguments (see [75], [91], and [94]) the $S$-compact set $\mathbf F=F^{(1)}\subset{\mathfrak R}_2(w)$ must be such that $\pi_2(\mathbf F)$ is mirror symmetric relative to the real line, while the second $S$-compact set ${\mathbf E}$ must be such that $\pi_2({\mathbf E})=E={E}_1\sqcup {E}_2$. Note that, by contrast to $\mathbf F\subset{\mathfrak R}_2(w)$, ${\mathbf E}\subset{\mathfrak R}_2(w)$ consists of two closed arcs ${\mathbf E}_1$ and ${\mathbf E}_2$ passing through the points $e_1$, $e_2$ and $e_3$, $e_4$, respectively. In addition, $\pi_2({\mathbf E}_j)=E_j$, $j=1,2$. These ‘general’ heuristic arguments are fully substantiated by numerical results. In Fig. 5 we present the results of numerical experiments for the function of the form (160), where
$$
\begin{equation*}
p=2,\quad A_1=\overline{A}_2\notin\mathbb{R}\quad\text{and}\quad q=3,\quad B_1=\overline{B}_2\notin\mathbb{R},\quad B_3\in\mathbb{R}
\end{equation*}
\notag
$$
(we do not give here the precise values of the parameters). Blue, red, and black dots are zeros of the Hermite–Padé polynomials of the first type $Q_{350,0}$, $Q_{350,1}$, and $Q_{350,2}$, respectively. They model the compact set $\pi_2(\mathbf{F})$. Note that the component of the complement to $\mathbf F$ on the Riemann surface ${\mathfrak R}(f)$ of $f$ that contains the point ${\mathbf z}=\infty^{(0)}$ is the Nuttall domain. It is the domain $\mathfrak D_1$ in the sense of Stahl [84] (also see § 5.1). Light blue points are zeros of the Hermite–Padé polynomial of the second type $P_{200,0}$, which model the compact set $\pi_2(\mathbf{E})$. In this case $\pi_2(\mathbf{E})=E_1 \sqcup E_2$ consists of two line segments. The component of the complement to ${\mathbf E}$ on ${\mathfrak R}(f)$ that contains the points ${\mathbf z}=\infty^{(0)}$ is the Stahl domain $\mathfrak D_0$ in the sense of [84] (also see § 5.1). The open set $\mathfrak D_1\setminus\overline{\mathfrak D}_0$ is the first sheet (here it is disconnected, so it is not a domain). The open set $\widehat{\mathbb{C}}\setminus\pi_2(\mathbf F)$ also consists of two domains, so it is not itself a domain. In this case the number of Chebotarev points of density zero on $F=\pi_2(\mathbf F)$ is five. We must point out that in Examples 1–4 above the compact set $F=\pi_2(\mathbf F)$ is disjoint from $E=\pi_2({\mathbf E})$. This is a heuristic law, always fulfilled in the class $\mathscr Z({E})$ (also see [76], Proposition 6) and even in the more general class of functions of the form (160). Example 5. In this example we consider another class of multivalued analytic functions, not based on the inverse Joukowsky function. For certain reasons (see [12]), instead of $z=\infty$, it is convenient to consider the analytic element $f_0(t)$ of such a function at the point $t=0$. We present this example to discuss a result of Stahl’s ([84], Theorem 4.3) on the limit distribution of the zeros of Hermite–Padé polynomials. Let $f$ be a multivalued analytic function defined by the explicit representation
$$
\begin{equation}
f(t):=\bigl((1-a_1t)(1-a_2t)(1-a_3t)(1-a_4t)\bigr)^{1/4},
\end{equation}
\tag{161}
$$
where all the $a_j$ lie in $\mathbb C\setminus\{0\}$. Thus, $f$ belongs to the class $\mathscr L$ of Laguerre multivalued analytic functions; see [61], [59], and [12]. The function (161) is an algebraic function of degree four. Hence, if we look at Hermite–Padé polynomials of the first type for the system of four functions $[1,f,f^2,f^3]$, then in accordance with [51], all these four polynomials have the same limit distribution of zeros. The situation changes17 drastically when we look at Hermite–Padé polynomials of the first type for the system of three function $[1,f,f^2]$. Namely, as Figs. 6–9 show, the numerical distribution of the zeros of $Q_{1000,1}$ is significantly distinct from the distribution of the zeros of $Q_{1000,0}$ and $Q_{1000,2}$ (on the other hand it is quite natural that the distributions of the zeros $Q_{1000,0}$ and $Q_{1000,2}$ coincide because the replacement of $f$ by $1/f$ does not take us out of the Laguerre class). In the case under consideration here we can see an unusual phenomenon, like nothing occurring for Padé polynomials or Hermite–Padé polynomials of the first type for a system $[1,f,\dots,f^{m-1}]$ when $f$ is an algebraic function of order $m$ (see [51]). This is because the asymptotic properties of Hermite–Padé polynomials for a system $[1,f,\dots,f^{m-1}]$ are determined by the $m$-sheeted Riemann surface associated (in the sense of Nuttall) with this system. This is perfectly similar to the fact that the asymptotic properties of Padé polynomials are determined by the two-sheeted hyperelliptic Riemann $\mathscr S_2(f)$ associated with the original multivalued function $f$ in the sense of Stahl: see [67], [66], [72], [13], [7], and [6]. In the present case, for a degree-four algebraic function defined by (161) the cases of the systems $[1,f,f^2,f^3]$ and $[1,f,f^2]$ are crucially different. Namely, in the first case the four-sheeted Riemann surface $\mathscr N_4(f)$ associated with18 $f$ in the sense of Nuttall coincides with the Riemann surface ${\mathfrak R}_4(f)$ of the function itself, so that $f$ is a single-valued meromorphic function on $\mathscr N_4(f)$. In the second case the three-sheetd Riemann surface $\mathscr N_3(f)$ associated with $f$ in the sense of Nuttall is distinct from the Riemann surface ${\mathfrak R}_4(f)$ of the function itself, so that $f$ is a single-valued meromorphic function in the two-sheeted Nuttall domain $\mathscr D_2(f)$ on $\mathscr N_3(f)$, but $f$ is multivalued on the whole of $\mathscr N_3(f)$.  Figure 6. The zeros of Hermite–Padé polynomials of the first type $Q_{n,0}(t)$ and $Q_{n,2}(t)$ of degree $n=1000$ for the system $[1,f_0(t),f_0^2(t)]$ (blue dots for $Q_{n,0}(t)$ and black dots for $Q_{n,2}(t)$; cf. Fig. 9). This set of blue and black dots approximates the second plate $F=\pi_2(\mathbf{F})$ of the Nuttall condenser.  Figure 7.The zeros of the Hermite–Padé polynomial of the second type $P_{2n,0}(t)$ of degree $2n=1000$ (light blue dots) corresponding to the pair $f_0(t)$, $f_0^2(t)$. The corresponding compact set $E$ is the projection $E=\pi_2(\mathbf{E})$ of the boundary between the zeroth and first sheets of the three-sheeted Riemann surface $\mathfrak{R}_3(f_0)$ associated in the sense of Nuttall with the given analytic element $f_0$; cf. Fig. 9.  Figure 8.The zeros of the Hermite–Padé polynomials of the first type $Q_{n,0}(t)$ and $Q_{n,2}(t)$ of degree $n=1000$ (blue dots for $Q_{n,0}(t)$ and black dots for $Q_{n,2}(t)$) and the zeros of the Hermite–Padé polynomial of the second type $P_{2n,0}(t)$ of degree $2n=1000$ (light blue dots). The corresponding pair of compact sets $E$ and $F$ forms the Nuttall condenser $(E,F)$. The plates of the condenser intersect in a finite number of points; cf. Figs. 2–5.  Figure 9.The zeros of the Hermite–Padé polynomial of the first type $Q_{n,1}(t)$ of degree $n=1000$ (red dots). We can see that the corresponding compact set is not a continuum; cf. Figs. 6 and 7.  Figure 10.The zeros of the Padé polynomials $\operatorname{PA}_{n,0}(t)$ and $\operatorname{PA}_{n,1}(t)$ of degree $n=500$ (blue and red dots, respectively) for the analytic element $f_0(t)$ of the function (161). The numerical distribution of these zeros is in full conformity with Stahl’s general theorem; see [85] and cf. Fig. 9. We select an analytic element $f_0\in{\mathscr H}(0)$ of $f$ at $t=0$ by the condition $f_0(0)=1$. For this element $f_0\in{\mathscr H}(0)$ of the function $f$ explicitly defined by (161) and any $n\in\mathbb{N}$, let $\operatorname{PA}_{n,0},\operatorname{PA}_{n,1}\in \mathbb{P}_n\setminus\{0\}$ be the Padé polynomials of degree $n$ corresponding to the point $t=0$, so that
$$
\begin{equation}
(\operatorname{PA}_{n,0}+\operatorname{PA}_{n,1}f)(t)=O(t^{2n+1}),\qquad z\to0.
\end{equation}
\tag{162}
$$
In Fig. 10 we show the zeros of the polynomials $\operatorname{PA}_{n,0}$ and $\operatorname{PA}_{n,1}$ of degree $n= 500$. Their numerical distribution is in full conformity with Stahl’s theorem (see [85] and [87]). The corresponding set of 1000 (blue and red) points approximates the Stahl compact set of minimum capacity with respect to the point $t=0$. Making the transformation $z=1/t$, in the plane ${\mathbb{C}}_z$ we obtain a classical compact set of minimum capacity. In our case the Stahl compact set in the plane ${\mathbb{C}}_z$ is the Chebotarev continuum of minimum capacity. In a similar way, for each $n\in\mathbb{N}$ the Hermite–Padé polynomials of the first type (at the point $t=0$) $Q_{n,0}(t),Q_{n,1}(t),Q_{n,2}(t)\in\mathbb{P}_n$, $Q_{n,0}\not\equiv0$, of degree ${n}$ for the system19 of analytic elements $[1,f_0(t),f^2_0(t)]$ are defined by the relation
$$
\begin{equation}
(Q_{n,0}+Q_{n,1}f_0+Q_{n,2}f^2_0)(t)=O(t^{3n+2}),\qquad t\to0.
\end{equation}
\tag{163}
$$
In Fig. 6 we show the zeros of the Hermite–Padé polynomials of the first type $Q_{n,0}$ and $Q_{n,2}$ of degree $n=1000$. The corresponding set of $1000$ (blue and black) points approximates the plate $F$ of the Nuttall condenser.20 We see from Fig. 6 that the compact set21 $F$ divides the Riemann sphere $\widehat{\mathbb{C}}_t$ into two domains, the unbounded domain $D_1\ni\infty$ and the bounded one $D_2\not\ni\infty$. In Fig. 8 we show the zeros of Hermite–Padé polynomials of the first type $Q_{n,0}$ and $Q_{n,2}$ of degree $n=1000$ (blue and black dots) and the zeros of Hermite–Padé polynomials of the second type $P_{2n,0}(t)$ of degree $2n=1000$ (light blue dots). In this way both plates of the Nuttall condenser $\mathrm N=(E,F)$ are modelled. Note that in this case the plates of the condenser intersect in a finite number of points: apart from the branch points, these are two Chebotarev points of positive density. Finally, in Fig. 9 we show the zeros of the Hermite–Padé polynomial of the first type $Q_{n,1} (t)$ of degree $n=1000$ (red points). Clearly, the corresponding compact set is not a continuum (cf. Figs. 6 and 7), and the distribution of the zeros of $Q_{n,1}$ is different from that of the zeros of $Q_{n,0}$ and $Q_{n,2}$. Moreover, comparing Figs. 9 and 8 we see that, in general, the distribution of the zeros of $Q_{n,1}$ is concentrated on the Nuttall condenser. The reason for such — ostensibly unusual — behaviour of $Q_{n,1}(t)$ is as follows. It is easy to see that for $t\in D_1\ni\infty$ we have $f(t^{(1)})=-f(t^{(0)})$, while $f(t^{(1)})=if(t^{(0)})$ for $t\in D_2$. Thus, $f(t^{(0)})+f(t^{(1)})\equiv0$ for $t\in D_1\ni\infty$, and $f(t^{(0)})+f(t^{(1)})=(1+i)f(t^{(0)})\not\equiv0$ for $t\in D_2$. The results in [89], [51], [93], and [49] substantiate a natural conjecture:
$$
\begin{equation}
\frac{Q_{n,1}(t)}{Q_{n,2}(t)}\xrightarrow{\operatorname{cap}}- \bigl(f(t^{(0)})+f(t^{(1)})\bigr)\equiv0,\qquad t\in D_1,\quad n\to\infty,
\end{equation}
\tag{164}
$$
and
$$
\begin{equation}
\frac{Q_{n,1}(t)}{Q_{n,2}(t)}\xrightarrow{\operatorname{cap}}- \bigl(f(t^{(0)})+f(t^{(1)})\bigr)\not\equiv0, \qquad t\in D_2, \quad n \to\infty.
\end{equation}
\tag{165}
$$
Relations (164) and (165), in combination with the results in [96] on the interpolation properties of rational functions constructed from Hermite–Padé polynomials take us to the following conclusion. The rational function $Q_{n,1}(t)/Q_{n,2}(t)$ must interpolate the limit function $-\bigl(f(t^{(0)})+f(t^{(1)})\bigr)$ in the domains $D_1$ and $D_2$, points of interpolation must have a limit distribution, and the support of the corresponding measure must lie on the compact set $E$. Since $f(t^{(0)})+f(t^{(1)})\not\equiv0$ in $D_2$, there is no way to calculate the corresponding points of interpolation. However, $f(t^{(0)})+f(t^{(1)})\equiv0$ in $D_1$, so points of interpolation there must coincide with the zeros of $Q_{n,1}(t)$, which Figs. 8 and 9 fully support: some zeros of $Q_{1000,1}$ are close to the part of zeros of $P_{1000,0}$ occurring in $D_1$. Since $f(t^{(0)})+f(t^{(1)})\not\equiv0$ in $D_2$, the other zeros of $Q_{1000,1}$ are distributed similarly to the zeros of $Q_{1000,0}$ and $Q_{1000,2}$, but this only holds on some part of $F$, namely, the compact set $F\setminus\partial D_1$ consisting of four arcs, the ‘interior’ part of the boundary of $D_2$ (cf. [22]).
6. Final remarks and several conjectures6.1.Let $f\in\mathbb C(z,w)$, and let $w_\infty$ be the analytic element of the function $w$ that we fixed above and $f_\infty\in{\mathscr H}(\infty)$ be the corresponding element of $f$. For $n\in\mathbb{N}$ let $P_{2n,0},P_{2n,1},P_{2n,2}$, $\deg{P_{2n,j}}\leqslant {2n}$, $P_{2n,0}\not\equiv0$, be the Hermite–Padé polynomials of the second type of degree $2n$ for the pair of functions $f$, $f^2$, so that
$$
\begin{equation}
\begin{alignedat}{2} R_{n,1}(z)&:=(P_{2n,0}f_\infty-P_{2n,1})(z)= O\biggl(\frac{1}{z^{n+1}}\biggr),&\qquad z&\to\infty, \\ R_{n,2}(z)&:=(P_{2n,0}f_\infty^2-P_{2n,2})(z)= O\biggl(\frac{1}{z^{n+1}}\biggr),&\qquad z&\to\infty. \end{alignedat}
\end{equation}
\tag{166}
$$
By analogy with [51] it is natural to assume that the following result holds (cf. [65] and [49]). Conjecture 1. Let $f\in{\mathbb{C}}(z,w)$, and let $f_\infty\in{\mathscr H}(\infty)$ be the analytic element of $f$ corresponding to the element $w_\infty$ fixed before. Then, as $n\to\infty$, Note that, since $f \in\mathbb C(z,w)$ is, in general, complex valued on the real line, the methods developed so far in [41], [10], [53], [54], and [56] on the basis of the Gonchar–Rakhmanov method [39] cannot be used to prove (167)–(169) (in this connection also see [61], [9], [74], [81], and [11]). Were (168) established, it would follow from this relation, in combination with (70), that the multi-valued analytic function $f\in\mathbb C(z,w)$ can constructively be recovered by use of Hermite–Padé polynomials of the first and second type on two (Nuttall) sheets of the three-sheeted Riemann surface $\mathscr N_3(w_\infty)$ from the given element $f_\infty$; here, as before, we speak about ‘constructive recovery’ in the sense of Henrici [44], § 2. As concerns constrictive approximation, see also [19], [5], [20], [89], [51], [93], [81], [23], [46], [49], [103], [3], [109], and the bibliography in these papers. Note that in this section we have only looked so far at the analytic element $w_\infty=w_{\infty^{(0)}}$. The following questions are natural here. What if we start with the element $w_{\infty^{(1)}}$ in place of $w_{\infty^{(0)}}$ considered before? What is the three-sheeted Riemann surface $ \mathscr N_3(w_{\infty^{(1)}})$ associated with the element $w_{\infty^{(1)}}$ of the four-valued analytic function $w$ in the sense of Nuttal, and what can we say about the asymptotic behaviour of Hermite–Padé polynomials corresponding to this $w_{\infty^{(1)}}$ or, more generally, to the analytic element $f_{\infty^{(1)}}$? 6.2.The problem of the asymptotic behaviour of Padé and Hermite–Padé polynomials is traditionally split into two components, namely, the geometric and analytic ones. In this paper we assume straight away that the geometric component is trivial and both plates of the Nuttall condenser lie on the real line: $E,F\subset\mathbb{R}$ and $E\cap F=\varnothing$. On the other hand, since $f\in{\mathbb{C}}(z,w)$ is complex valued on the real line, the analytic component is no longer standard. Thus, the following step looks natural. Assume that in (63) we have a more general situation, when for some $j \in\{1,\dots,m\}$ we still have $A_j<B_j$, but for some $k\in\{1,\dots,m\}$ we have $A_k=\overline{B}_k\notin\mathbb{R}$. In this case we must use the general definition of the Nuttall condenser $\mathrm N=(E,F)$, just as Rakhmanov and this author [75] did in 2013. Namely, we have $E\subset\mathbb{R}$, but we cannot say the same about $F$. Instead, we assume that $F$ is mirror symmetric realtive to the real line, that is, $z\in F$ if and only if $\overline{z}\in F$. On the other hand we assume, as usual, that $E\cap F=\varnothing$. Conjecture 2. If $f\in{\mathbb{C}}(z,w)$ and $f_\infty\in{\mathscr H}(\infty)$, then under the above assumptions about the parameters $A_j$ and $B_j$ relations (69) –(70) and (167) –(169) hold as $n\to \infty$. 7. Applications7.1.Let $E:=\bigsqcup\limits_{j=1}^p E_j\subset{\mathbb{C}}$, where each of the disjoint sets $E_j$ is a continuum (not degenerate to a point) in ${\mathbb{C}}$ with connected complement $\widehat{\mathbb{C}}\setminus{E_j}$. In a similar way, let $F:=\bigsqcup\limits_{k=1}^q F_k\subset{\mathbb{C}}$ consist of a finite number of disjoint continua with connected complements. Also assume that $E\cap F=\varnothing$. Set
$$
\begin{equation*}
D:=\widehat{\mathbb{C}}\setminus{E}\quad\text{and}\quad \Omega:=\widehat{\mathbb{C}}\setminus F.
\end{equation*}
\notag
$$
Thus, $D$ and $\Omega$ are regular domains with respect to the Dirichlet problem, and we have $D\supset F$ and $\Omega\supset E$. Let $M_1(E)$ be the family of all (positive) Borel unit measures with support on $E$, and let $g_F(t,z)$, $z,t\in\Omega$, be the Green’s function for $\Omega$ with logarithmic singularity at $t=z$, $g_F(t,z)\equiv0$ for $t\in F$. Given $\mu\in M_1(E)$, consider
$$
\begin{equation}
V^\mu(z):=\int_E\log\frac{1}{|z-t|}\,d\mu(t)\quad\text{and}\quad G^\mu_F(z):=\int_E g_F(t,z)\,d\mu(t),
\end{equation}
\tag{170}
$$
the logarithmic and Green’s (with respect to $\Omega$) potentials of the measure $\mu$, respectively. Fix $\theta\in(0,\infty)$, and let be a mixed Green’s logarithmic kernel, be the corresponding mixed potential, and
$$
\begin{equation}
J_{\theta,F}(\mu):=\int_E P_{\theta,F}^\mu(z)\,d\mu(z)= \int\!\!\!\int_{E\times E}k_\theta(z,\zeta)\,d\mu(z)\,d\mu(\zeta)
\end{equation}
\tag{173}
$$
be the corresponding energy functional (‘energy’). Set Then the following result holds. Theorem 7. There exists a unique measure $\lambda_E\in M_1(E)$ with the following properties: Furthermore, properties (a) and (b) are equivalent. Proof. The existence of $\lambda_E\in M_1(E)$ with property (174) is a direct consequence of the principle of descent (see [52], Chap. I, § 3, Theorem 1.3, [26], and [27]). Part (a) is proved.
To prove part (b) of Theorem 7 we use the approach from [40], § 3.2 (the proof of Lemma 6), in order to prove first of all, using the fact that the mixed kernel (171) is positive (see [26] and [27]), that the minimizing measure $\lambda_E$ is unique. After that [40] the following equilibrium relations can be established:
$$
\begin{equation}
P^{\lambda_E}_{\theta,F}(z)\begin{cases} \equiv c_E(\theta),& z\in \operatorname{supp} \lambda_E, \\ \geqslant c_E(\theta), & z\in E\setminus\operatorname{supp}\lambda_E. \end{cases}
\end{equation}
\tag{176}
$$
Now let $\lambda_F=\lambda_F(\,\cdot\,;\theta):=\beta_F(\lambda_E)$, $\operatorname{supp}\lambda_F=F$, be the balayage of $\lambda_E$ from $\Omega$ to the boundary $\partial\Omega=F$. Then by the definition of balayage
$$
\begin{equation}
V^{\lambda_E}(z)-G^{\lambda_E}_F(z)\equiv V^{\lambda_F}(z)+\operatorname{const},\qquad z\in \widehat{\mathbb{C}},
\end{equation}
\tag{177}
$$
where the constant satisfies
$$
\begin{equation}
\operatorname{const}=-G_F^{\lambda_E}(\infty)= -\int_E g_F(\zeta,\infty)\,d\lambda_E(\zeta)=-c_0.
\end{equation}
\tag{178}
$$
Thus, it follows from (177) and the equilibrium relations (176) that
$$
\begin{equation}
(\theta+1)V^{\lambda_E}(z)\equiv V^{\lambda_F}(z)+c_E(\theta)-c_0,\qquad z\in\operatorname{supp}\lambda_E\subset E,
\end{equation}
\tag{179}
$$
where $\theta\geqslant0$ and $\operatorname{supp}\lambda_E\cap \operatorname{supp}\lambda_F=\varnothing$. Because $\theta\geqslant0$, it follows from (179) by the principle of domination (see [77], Chap. 2, Theorem 3.2 on p. 104) that
$$
\begin{equation}
(\theta+1)V^{\lambda_E}(z)\leqslant V^{\lambda_F}(z)+c_E(\theta)-c_0,\qquad z\in\widehat{\mathbb{C}}.
\end{equation}
\tag{180}
$$
We conclude from (176) and (180) that $P^{\lambda_E}_{\theta,F}(z)\leqslant c_E(\theta)$, and therefore $P^{\lambda_E}_{\theta,F}(z)\equiv c_E(\theta)$, $z\in E$, and $\operatorname{supp}\lambda_E=E$. $\Box$ Theorem (principle of domination; [77]). Let $\mu$ and $\nu$ be two positive finite Borel measures with compact support in the plane ${\mathbb{C}}$ such that the total mass of $\nu$ is not greater than the mass of $\mu$. Also assume that $\mu$ has a finite logarithmic energy. If for some constant $\operatorname{const}$ $\mu$-almost everywhere, then (181) holds for all $z\in{\mathbb{C}}$. 7.2.Let $M_1(F)$ denote the set of unit Borel measures $\nu$ with support on $F$. Let $g_E(\zeta,z)$ be the Green’s function for the domain $D=\widehat{\mathbb{C}}\setminus{E}$, and let be the Green’s potential of $\nu\in M_1(F)$ (with respect to $D$) and
$$
\begin{equation}
J_{\theta,E}(\nu):=\int_FP^\nu_{\theta,E}(\zeta)\,d\nu(\zeta)+ 2\theta\int_F g_E(\zeta,\infty)\,d\nu(\zeta)
\end{equation}
\tag{182}
$$
be the energy of $\nu\in M_1(E)$ with respect to the mixed Green’s logarithmic potential with external field $\theta g_E(z,\infty)$. Let $\nu_F\in M_1(F)$ be the (unique in $M_1(F)$) extremal measure for the energy functional with respect to the potential $P^\nu_{\theta,E}(z)$ with external field $\theta g_E(z,\infty)$:
$$
\begin{equation}
J_{\theta,E}(\nu_F)=\inf_{\nu\in M_1(F)}J_{\theta,E}(\nu).
\end{equation}
\tag{184}
$$
Then $\nu_F$, $\operatorname{supp}\nu_F=F$, is also the (unique) equilibrium measure on $F$ with respect to the potential $P^\nu_{\theta,E}(z)$ with external field $\theta g_E(z,\infty)$, that is,
$$
\begin{equation}
P^{\nu_F}_{\theta ,E}(z)+\theta g_E(z,\infty)\equiv c_F(\theta)= \operatorname{const}\quad\text{on}\ \ F.
\end{equation}
\tag{185}
$$
For $\mu \in M_1(E)$ let $\widetilde\mu$ denote the balayage of $\mu$ from $\Omega$ to $F$, that is, the unique measure in $M_1(F)$, such that
$$
\begin{equation}
V^{\widetilde\mu}(z)=V^\mu(z)-G_F^\mu(z)+c_F(\mu),\qquad z\in \widehat{\mathbb{C}},
\end{equation}
\tag{186}
$$
where $c_F(\mu)=G_F^\mu(\infty)$. Then the following lemma holds. Lemma 1. Let $F$ be, as above, a regular compact set with respect to the Dirichlet problem such that $F\cap E=\varnothing$, and let $\lambda=\lambda_E\in M_1(E)$ be the extremal measure for the energy functional (173). Then $\widetilde\lambda\in M_1(F)$ is the extremal measure for the energy functional (182). In addition, where $\gamma_E$ is the Robin constant for $E$. Proof. For an arbitrary measure $\mu\in M_1(E)$, replacing the first term in the sum $G_F^\mu(z)+G^{\widetilde\mu}_E(z)$ by its expression with the use of (186), immediately from the definitions of $V^{\widetilde\mu}$ and $G^{\widetilde\mu}_E$ we obtain
$$
\begin{equation*}
\begin{aligned} \, G_F^\mu(z)+G^{\widetilde\mu}_E(z)&=V^\mu (z)-V^{\widetilde\mu}(z)+ G^{\mu}_F(\infty)+G^{\widetilde\mu}_E(z) \\ &=V^\mu(z)+G^{\mu}_F(\infty)+\int_F\bigr(\log|z-t|+ g_E(t,z)\bigr)\,d\widetilde\mu(t). \end{aligned}
\end{equation*}
\notag
$$
It follows directly from the definition of the Green’s function $g_E(t,z)$ that for all $t\in F$ the integrand in the above integral is a harmonic functions of $z\in\widehat{\mathbb{C}}\setminus E$. Hence the function
$$
\begin{equation}
\begin{aligned} \, H^\mu_{\theta,F}(z)&:=P^\mu_{\theta,F}(z)+G^{\widetilde\mu}_E(z)+ \theta g_E(z,\infty) \nonumber \\ &=\theta V^\mu (z)+G_F^\mu(z)+G^{\widetilde\mu}_E(z)+\theta g_E(z,\infty) \end{aligned}
\end{equation}
\tag{189}
$$
is harmonic in $\widehat{\mathbb{C}}\setminus E$. It is easy to see that $H^\mu_{\theta,F}(z)\big|_E=P^\mu_{\theta,F}(z)\big|_E$. Hence for $\mu=\lambda$, where $\lambda$ is the extremal — and therefore equlibrium — measure on $E$ with respect to the potential $P^\mu_{\theta,F}$, the function $H^\lambda_{\theta,F}$, which is harmonic in $\widehat{\mathbb{C}}\setminus E$, takes the constant value on $E$, and therefore it is a constant, namely,
Replacing the first term in this equality by its expression from (186) we obtain
$$
\begin{equation}
\theta V^{\widetilde\lambda}(z)+(1+\theta)G_F^\lambda(z)+ G_E^{\widetilde\lambda}(z)+\theta g_E(z,\infty)\equiv c_F(\theta),\qquad z\in \widehat{\mathbb{C}},
\end{equation}
\tag{192}
$$
where
Since $G_F^\lambda(z)=0$ on $F$, equality (192) means that the measure $\widetilde\lambda\in M_1(F)$ has the equilibrium property on $F$ with respect to the potential $P^{\widetilde\lambda}_{\theta,E}(z)$ with external field $\theta g_E(z,\infty)$:
$$
\begin{equation}
\begin{aligned} \, P^{\widetilde\lambda}_{\theta,E}(z)+\theta g_E(z,\infty)&= \theta V^{\widetilde\lambda}(z)+G_E^{\widetilde\lambda}(z)+ \theta g_E(z,\infty) \nonumber \\ &\equiv c_F(\theta)=\operatorname{const},\qquad z\in F. \end{aligned}
\end{equation}
\tag{194}
$$
It is well known [40], [38] that the measure with equilibrium property is unique in $M_1(F)$. Hence $\widetilde\lambda=\lambda_F$. Thus, we have shown that if $\lambda_E\in M_1(E)$ is the equilibrium measure for $P^\mu_{\theta,F}$, defined by (172), then the equilibrium measure $\lambda_F\in M_1(F)$ for the potential $P^\nu_{\theta,E}$, defined by (183), with external field $\theta g_E(z,\infty)$ coincides with the balayage of $\lambda_E$ to $F$.
Now we prove (188). It follows from the definition (182) of the energy functional $J_{\theta,E}(\tilde{\lambda})$, equality (194), the definition of $G_E^{\widetilde\lambda}(z)$ at $z=\infty$, and (193) that
$$
\begin{equation}
\begin{aligned} \, J_{\theta,E}(\widetilde\lambda)&= \int_F\bigl(P^{\widetilde\lambda}_{\theta,E}(z)+\theta g_E(z,\infty)+ \theta g_E(z,\infty)\bigr)\,d\widetilde\lambda (z) \nonumber \\ &=c_F(\theta)+\theta G_E^{\widetilde\lambda}(\infty)= c_E(\theta)+\theta G_F^{\lambda}(\infty)+ \theta G_E^{\widetilde\lambda}(\infty). \end{aligned}
\end{equation}
\tag{195}
$$
From identity (191) for $z=\infty$ we obtain
$$
\begin{equation*}
\theta\gamma_E+G_F^{\lambda}(\infty)+G_E^{\widetilde\lambda}(\infty)= c_E(\theta).
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation}
J_{\theta,E}(\widetilde\lambda)= c_E(\theta)+\theta (c_E(\theta)-\theta\gamma_E)= (1+\theta)c_E(\theta)-\theta^2\gamma_E.
\end{equation}
\tag{196}
$$
Since the definition (173) of $J_{\theta,F}(\lambda)$ and (190) yield the equality
$$
\begin{equation*}
J_{\theta,F}(\lambda)=\int_EP^{\lambda}_{\theta,F}(z)\,d\lambda(z)= c_E(\theta),
\end{equation*}
\notag
$$
the above equality (196) coincides with (188). $\Box$ Remark 1. The scalar equilibrium problem (16) considered here is equivalent to the following vector problem (cf. [40], [38], and [4]): where $\lambda_1=\lambda_E\in M_1(E)$, $\lambda_2=\lambda_F\in M_1(F)$, and $\vec\lambda=(\lambda_1,\lambda_2)$ is the vector equilibrium measure. 7.3.As before, let $\varphi(z)=z+(z^2-1)^{1/2}\sim 2z$ as $z\to\infty$. Lemma 2. Let $E=[-1,1]$, $F\subset\mathbb{R}$, $E\cap F=\varnothing$, let $\mu$ be a measure in $M_1(F)$, and let Then where $\beta_E(\mu)$ is the balayage of $\mu$ from the domain $D=\widehat{\mathbb{C}}\setminus E$ to its boundary $E$ and $\tau_E$ is the Chebyshev measure on the interval $E$. Proof. In fact, as both $E$ and $F$ lie on the real line $\mathbb{R}$ and since $\varphi(z)$ takes real values for $t\in\mathbb{R}\setminus E$, for the Green’s function $g_E(t,z)$ of $D$ we have
$$
\begin{equation}
g_E(t,z)=\log\frac{|1-\varphi(z)\overline{\varphi(t)}|} {|\varphi(z)-\varphi(t)|}=\log\frac{|1-\varphi(z)\varphi(t)|} {|\varphi(z)-\varphi(t)|}
\end{equation}
\tag{198}
$$
for $z,t\in\mathbb{R}\setminus E$. Now we use the following identity (see the definition of $\Phi({\mathbf z})$ in § 2.2):
$$
\begin{equation*}
z-a\equiv -\frac{(\Phi({\mathbf z})-\Phi({\mathbf a})) (1-\Phi({\mathbf z})\Phi({\mathbf a}))} {2\Phi({\mathbf z})\Phi({\mathbf a})}\,,\qquad z,a\in D,
\end{equation*}
\notag
$$
which is easy to verify (see [43] and [90]). Using this relation, representation (198), and the link between $\Phi({\mathbf z})$ and $\varphi(z)$, for $t\in\mathbb{R}\setminus E$ we finally obtain
$$
\begin{equation}
g_E(t,z)=\log\frac{|1-\varphi(z)\varphi(t)|^2} {2|z-t|\cdot|\varphi(z)\varphi(t)|}\,.
\end{equation}
\tag{199}
$$
Now, in accordance with identity (199), for the Green’s potential $G^\mu_E(z)$ of $\mu\in M_1(F)$ we have
$$
\begin{equation}
\begin{aligned} \, G^\mu_E(z)&:=\int_F g_E(t,z)\,d\mu(t) \nonumber \\ &=2\int_F\log|1-\varphi(z)\varphi(t)|\,d\mu(t)+V^\mu(z)- \log|\varphi(z)|+\operatorname{const} \nonumber \\ &=2v(z;\mu)+V^\mu(z)-\gamma_E+V^{\tau_E}(z)+\operatorname{const}. \end{aligned}
\end{equation}
\tag{200}
$$
Next we apply the operator $\mathrm{dd}^\mathrm{c}$ to $v(z;\mu)$. Then (see [25])
$$
\begin{equation*}
\frac{1}{2\pi}\,\mathrm{dd}^\mathrm{c} G^\mu_E(z)=\beta_E(\mu)-\mu,
\end{equation*}
\notag
$$
where $\beta_E(\mu)$ is the balayage of $\mu$ from $D$ to $E$. Hence using (200) we arrive at the relation
$$
\begin{equation*}
2\pi(\beta_E(\mu)-\mu)=2\mathrm{dd}^\mathrm{c} v(z;\mu)+2\pi(\delta_\infty-\mu)+ 2\pi(\delta_\infty-\tau_E).
\end{equation*}
\notag
$$
Thus, Therefore,
Citation in format AMSBIB
\Bibitem{Sue25} |
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