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Article overview
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On Polya' Theorem in Several Complex Variables | Ozan Günyüz
; Vyacheslav Zakharyuta
; | Date: |
1 Sep 2016 | Abstract: | Let $K$ be a compact set in $mathbb{C}$, $f$ a function analytic in
$overline{mathbb{C}}smallsetminus K$ vanishing at $infty $. Let $% fleft(
z
ight) =sum_{k=0}^{infty }a_{k} z^{-k-1}$ be its Taylor expansion at
$infty $, and $H_{s}left( f
ight) =det left( a_{k+l}
ight) _{k,l=0}^{s}$
the sequence of Hankel determinants. The classical Polya inequality says that
[ limsuplimits_{s
ightarrow infty }leftvert H_{s}left( f
ight)
ightvert ^{1/s^{2}}leq dleft( K
ight) , ]% where $dleft( K
ight) $ 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. | Source: | arXiv, 1609.0218 | Services: | Forum | Review | PDF | Favorites |
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