The Internal Polya Inequality for $\mathbb{C}$-convex Domains in $\mathbb{C}^n$
Ozan Günyüz, Vyacheslav Zakharyuta
TL;DR
The paper develops multivariate Polya-type inequalities for $\mathbb{C}$-convex domains in $\mathbb{C}^{n}$ by embedding the problem in the framework of Hankel-type determinants built from Taylor coefficients of functions in $A(D)$ at a point $a$. It first recalls the classical and multivariate Polya results, then introduces internal analogs with $s$-indicatrices and weighted determinants, proving an internal Polya inequality (Theorem ipt) for strictly linearly convex domains via reduction to Zakharyuta's outer inequality; it also discusses the necessity of weights and provides counterexamples to prior claims. The paper further defines and analyzes the internal transfinite diameter $d(c,\partial D)$, contrasts it with $d((D-a)^{*})$, and presents explicit calculations for balls and polydiscs, along with open problems on nonweighted internal Polya bounds and a conjecture linking $\rho$ to $d(c,\partial D)$ for favorable domains. Overall, it clarifies how domain geometry and weighting interact in multivariate Polya-type bounds and introduces a formal framework for internal transfinite diameters in this setting.
Abstract
Let $K\subset \mathbb{C}$ be a polynomially convex compact set, $f$ be a function analytic in a domain $\overline{\mathbb{C}}\smallsetminus K$ with Taylor expansion $f\left( z\right) =\sum_{k=0}^{\infty }\frac{a_{k}}{z^{k+1}} $ at $\infty $, and $H_{s}\left( f\right) :=\det \left( a_{k+l}\right) _{k,l=0}^{s}$ related Hankel determinants. The classical Polya theorem \cite% {P} says that \[ \limsup_{s\rightarrow \infty }\left\vert H_{s}\left( f\right) \right\vert ^{1/s^{2}}\leq d\left( K\right) , \]% where $d\left( K\right) $ is the transfinite diameter of $K$. The main result of this paper is multivariate internal analogs of Polya's inequality for $\mathbb{C}$-convex (=strictly linearly convex) domains $D\subset \mathbb{C}^{n}$ and weighted Hankel-type determinants, constructed from the Taylor coefficients of a function $f\in A\left( D\right) $ at a given point $% a\in D$; therewith the weights are generated by $s$-indicatrices of the sequence of analytic functionals biorthogonal to the system of monomials in $% \mathbb{C}^{n}$. It is proved by the reduction to the outer multivariate analog of Polya's inequality (Zakharyuta, Math. USSR Sbornik, \textbf{25 }% (1975)) and is based on the characterization of the strict linear convexity in terms of $s$-indicatrices (S. Znamenskii, Siberian Math. J. \textbf{26 } (1985)).