On Polya' Theorem in Several Complex Variables

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.