On Pólya's Theorem in several complex variables

Ozan Günyüz; Vyacheslav Zakharyuta

Banach Center Publications, Tome 104 (2015), p. 149-157 / Harvested from The Polish Digital Mathematics Library

Résumé

Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f (z) = \sum_{k = 0}^{\infty} a_{k} z^{- k - 1}$ be its Taylor expansion at ∞, and $H_{s} (f) = d e t {(a_{k + l})}_{k, l = 0}^{s}$ the sequence of Hankel determinants. The classical Pólya inequality says that $l i m s u p_{s \to \infty} {| H_{s} (f) |}^{1 / s ²} \leq d (K)$ , 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.

Publié le : 2015-01-01

Zbl 06556708

EUDML-ID : urn:eudml:doc:281663

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     author = {Ozan G\"uny\"uz and Vyacheslav Zakharyuta},
     title = {On P\'olya's Theorem in several complex variables},
     journal = {Banach Center Publications},
     volume = {104},
     year = {2015},
     pages = {149-157},
     zbl = {06556708},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc107-0-10}
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Ozan Günyüz; Vyacheslav Zakharyuta. On Pólya's Theorem in several complex variables. Banach Center Publications, Tome 104 (2015) pp. 149-157. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc107-0-10/