Growth and zero sets of analytic families of Cauchy-Stieltjes integrals
type: Item
instance of: scholarly article
title: Growth and zero sets of analytic families of Cauchy-Stieltjes integrals
zbMATH Open document ID: 0793.30035
DOI: 10.1007/BF02788843
author: Thomas H. MacGregor
author: David J. Hallenbeck
published in: Journal d'Analyse Mathématique
publication date: 1994-08-15T00:00:00Z
review text: Let \(X\) be the boundary of the unit disk \(D\). The authors continue their study of functions \(f(z)\), \(z\in D\) which can be represented as \[ f(z)=\int_ X (1-xz)^{-\alpha} d\mu(x), \] where \(\mu\) is a probability measure and \(\alpha>0\). (A slightly different formula works for \(\alpha=0\).) The functions so representable form \({\mathcal F}_ \alpha\). For functions in a given \({\mathcal F}_ \alpha\) the authors determine the corresponding Hardy and Bergman spaces to which they belong. The extremal function \(f(z)= (1-z)^{-\alpha}\) shows what to expect. When \(\alpha\leq 1\), the zeros satisfy a Blaschke condition while if \(\alpha>1\) there is a weaker condition: \(\prod^ n(1/| a_ k|= o(n^{\alpha-1})\), but neither is sufficient. However, the Blaschke condition remains sharp when the zeros are confined to a single ray. (This is related to earlier work of \textit{F. Bagemihl}, \textit{P. Erdős} and \textit{W. Seidel} [Ann. Sci. École Norm. Super., III. Sér. 70, 133- 147 (1953; Zbl 0053.238)].) An extensive study is made of radial growth as \(r\to 1\). This is of significance only when \(\alpha>1\). An interesting device (p.244) is a representation of \(f\in{\mathcal F}_ \alpha\) in terms of a function \(g\in{\mathcal F}_ 1\), valid on any concentric disk.
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 528675
zbMATH Keywords: Cauchy-Stieltjes integrals
zbMATH Keywords: zero sets
zbMATH Keywords: Bergman spaces
reviewed by: David Drasin
MaRDI profile type: Publication
cites work: Q4177796
cites work: On the zeros of functions with bounded Dirichlet integrals
cites work: Sur quelques propriétés frontières des fonctions holomorphes définies par certains produits dans le cercle-unité
cites work: Q5736202
cites work: Q5618011
cites work: Q3704135
cites work: Radial growth and variation of bounded analytic functions
cites work: On Functions with Positive Real Part
cites work: A criterical growth rate for functions regular in a disk
cites work: Closure Properties of Families of Cauchy-Stieltjes Transforms
cites work: Multipliers of Families of Cauchy-Stieltjes Transforms
cites work: Bounded analytic families of Cauchy-Stieltjes integrals
cites work: Univalent functions with restricted growth and cauchy-stieltjes integrals
cites work: Zeros of functions in the Bergman spaces
cites work: Q5644244
cites work: On the Derivative of a Univalent Function
cites work: Q3786690
cites work: On the coefficients of close-to-convex functions
cites work: Zeros of Lipschitz functions analytic in the unit disc
cites work: Q3259806
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description: scientific article; zbMATH DE number 528675