On Polya' Theorem in Several Complex Variables

Abstract

Let $K$ be a compact set in $\mathbb{C}$, $f$ a function analytic in $\overline{\mathbb{C}}\setminus K$ vanishing at $\infty$. Let 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.

9 pages. arXiv admin note: substantial text overlap with arXiv:1605.09314