# On Pólya's Theorem in several complex variables

Ozan Günyüz; Vyacheslav Zakharyuta

Banach Center Publications (2015)

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

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topOzan 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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