On Pólya’s Theorem in several complex variables
Volume 107 / 2015
Banach Center Publications 107 (2015), 149-157
MSC: Primary: 32A22, 32A70, 32U35; Secondary: 46E10.
DOI: 10.4064/bc107-0-10
Abstract
Let $K$ be a compact set in $\mathbb{C}$, $f$ a function analytic in $\overline{\mathbb{C}}\setminus K$ vanishing at $\infty $. Let $f( z) =\sum_{k=0}^{\infty }a_{k}z^{-k-1}$ be its Taylor expansion at $\infty $, and $H_{s}( f) =\det (a_{k+l}) _{k,l=0}^{s}$ the sequence of Hankel determinants. The classical Pólya inequality says that \[ \limsup_{s\rightarrow \infty }\left\vert H_{s}( f)\right\vert ^{1/s^{2}}\leq d( K) , \] where $d( K)$ 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 Pólya’s inequality, considered by the second author in Math. USSR Sbornik 25 (1975), 350–364.
