Remarks on Hardy-Orlicz classes
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)].
scientific article; zbMATH DE number 4073383