Remarks on Hardy-Orlicz classes
type: Item
instance of: scholarly article
title: Remarks on Hardy-Orlicz classes
zbMATH Open document ID: 0657.30026
DOI: 10.1007/BF01198631
author: Morisuke Hasumi
author: Saburô Kataoka
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let \(\Phi\) be a nonconstant, nondecreasing convex function defined in R with \(\lim_{t\to -\infty}\Phi (t)=0\) and let \(H^{\Phi}\) be the associated Hardy-Orlicz class, i.e., the set of all analytic functions in the open unit disk D such that the subharmonic function \(\Phi\) (log\(| f|)\) has a harmonic majorant on D. The main purpose of the paper under review is to present conditions on \(\Phi\) to guarantee that the set of Schur multipliers of \(H^{\Phi}\) contains (respectively is contained in) the set \(H^{\infty}\) of all bounded analytic functions in D. Moreover, the authors obtain the fact that \(H^{\Psi}\subseteq H^{\Phi}\) if and only if \(\limsup_{t\to \infty}\Phi (t)/\Psi (t)<\infty\). Finally, they show how their theorems improve some of the results of \textit{W. Deeb}, \textit{R. Khalil} and \textit{M. Marzuq} [Bull. Austr. Math. Soc. 34, 177-189 (1986; Zbl 0589.46019)].
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 4073383
zbMATH Keywords: Hardy-Orlicz class
zbMATH Keywords: Schur multipliers
reviewed by: Raymond Mortini
MaRDI profile type: Publication
cites work: H(ϕ) Spaces
cites work: Isometric multiplication of Hardy-Orlicz spaces
cites work: Q5618011
cites work: Abstract analytic function theory and Hardy algebras
full work available at URL: https://doi.org/10.1007/bf01198631
OpenAlex ID: W2063794043
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description: scientific article; zbMATH DE number 4073383