On Polya' Theorem in Several Complex Variables

TL;DR

This paper extends Polya's inequality from one complex variable to several complex variables, establishing sharpness of the multivariate inequality for certain classes of compact sets.

Contribution

It proves a sharpness result for the multivariate Polya inequality, generalizing the classical inequality to higher dimensions.

Findings

Established sharpness of the multivariate Polya inequality

Extended classical inequality to several complex variables

Provided conditions under which the inequality is sharp

Abstract

Let $K$ be a compact set in $\mathbb{C}$, $f$ a function analytic in $\overline{\mathbb{C}}\smallsetminus K$ vanishing at $\infty $. Let $% f\left( z\right) =\sum_{k=0}^{\infty }a_{k}\ z^{-k-1}$ be its Taylor expansion at $\infty $, and $H_{s}\left( f\right) =\det \left( a_{k+l}\right) _{k,l=0}^{s}$ the sequence of Hankel determinants. The classical Polya inequality says that \[ \limsup\limits_{s\rightarrow \infty }\left\vert H_{s}\left( f\right) \right\vert ^{1/s^{2}}\leq d\left( K\right) , \]% where $d\left( K\right) $ is the transfinite diameter of $K$. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Polya's inequality, considered by the second author in Math. USSR Sbornik, 25 (1975), 350-364.