# On Pólya's Theorem in several complex variables

• Volume: 107, Issue: 1, page 149-157
• ISSN: 0137-6934

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## Abstract

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Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f\left(z\right)={\sum }_{k=0}^{\infty }{a}_{k}{z}^{-k-1}$ be its Taylor expansion at ∞, and ${H}_{s}\left(f\right)=det{\left({a}_{k+l}\right)}_{k,l=0}^{s}$ the sequence of Hankel determinants. The classical Pólya inequality says that $limsu{p}_{s\to \infty }{|{H}_{s}\left(f\right)|}^{1/s²}\le d\left(K\right)$, 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.

## How to cite

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Ozan Günyüz, and Vyacheslav Zakharyuta. "On Pólya's Theorem in several complex variables." Banach Center Publications 107.1 (2015): 149-157. <http://eudml.org/doc/281663>.

@article{OzanGünyüz2015,
abstract = {Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f(z) = ∑_\{k=0\}^\{∞\} a_\{k\}z^\{-k-1\}$ be its Taylor expansion at ∞, and $H_\{s\}(f) = det(a_\{k+l\})_\{k,l=0\}^\{s\}$ the sequence of Hankel determinants. The classical Pólya inequality says that $lim sup_\{s→∞\} |H_\{s\}(f)|^\{1/s²\} ≤ d(K)$, 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.},
author = {Ozan Günyüz, Vyacheslav Zakharyuta},
journal = {Banach Center Publications},
language = {eng},
number = {1},
pages = {149-157},
title = {On Pólya's Theorem in several complex variables},
url = {http://eudml.org/doc/281663},
volume = {107},
year = {2015},
}

TY - JOUR
AU - Ozan Günyüz
AU - Vyacheslav Zakharyuta
TI - On Pólya's Theorem in several complex variables
JO - Banach Center Publications
PY - 2015
VL - 107
IS - 1
SP - 149
EP - 157
AB - Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f(z) = ∑_{k=0}^{∞} a_{k}z^{-k-1}$ be its Taylor expansion at ∞, and $H_{s}(f) = det(a_{k+l})_{k,l=0}^{s}$ the sequence of Hankel determinants. The classical Pólya inequality says that $lim sup_{s→∞} |H_{s}(f)|^{1/s²} ≤ d(K)$, 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.
LA - eng
UR - http://eudml.org/doc/281663
ER -

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