On a theorem of Korenblum
type: Item
instance of: scholarly article
title: On a theorem of Korenblum
zbMATH Open document ID: 0812.30003
DOI: 10.1007/BF02559530
published in: Arkiv för Matematik
publication date: 1994-07-12T00:00:00Z
review text: \(A^{-\alpha}\) \((\alpha> 0)\) is the set of functions \(f\) analytic in the unit disk \(U= \{z: | z|< 1\}\) satisfying \[ \sup_{z\in U} (1- | z|)^ \alpha | f(z)|< \infty. \] For a given sequence \(Z\) from \(U\) we denote by \(\rho(Z)\) the infimum of those \(\alpha\) such that \(Z\) is the sequence of zeros of a function in \(A^{-\alpha}\). It is proved that \(\rho(Z)\) can be expressed in terms of a certain density of \(Z\). This notion of density was introduced by \textit{B. Korenblum} [Acta Math. 135, 187-219 (1975; Zbl 0323.30030)], who proved a less precise result. Some fine estimates of \textit{E. J. Specht} [see Trans. Am. Math. Soc. 71, 183-196 (1951; Zbl 0043.302)] on the mapping function and its derivative in conformal mapping of so called nearly circular regions play a crucial role in the proof.
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 605124
author: Kristian Seip
reviewed by: Kristian Seip
MaRDI profile type: Publication
cites work: On the closure of characters and the zeros of entire functions
cites work: Zeros of functions in the Bergman spaces
cites work: An extension of the Nevanlinna theory
cites work: Q4000499
cites work: Beurling type density theorems in the unit disk
cites work: Estimates on the Mapping Function and Its Derivatives in Conformal Mapping of Nearly Circular Regions
description: scientific article; zbMATH DE number 605124