Integral transforms of functions with the derivative in a halfplane
type: Item
instance of: scholarly article
title: Integral transforms of functions with the derivative in a halfplane
zbMATH Open document ID: 0943.30010
DOI: 10.1007/BF02785576
author: Saminathan Ponnusamy
author: Frode Rønning
published in: Israel Journal of Mathematics
publication date: 2000-09-05T00:00:00Z
review text: Let \({\mathcal A}\) denote the class of functions \(f\) holomorphic in the unit disc \(\Delta\) with normalization \(f(0)= 0\), \(f'(0)= 1\). Denote by \({\mathcal K}\), \({\mathcal S}_\gamma^*\) and \({\mathcal S}\) the subclasses of \({\mathcal A}\) containing functions that are convex, starlike of order \(\gamma\) and univalent, respectively. For a function \(f\in{\mathcal A}\) we define the known integral transform \(F_c(z)= (c+1) \int_0^1 t^{c-1} f(tz) dt\), \(c>-1\). The authors define the classes \[ {\mathcal P}_\beta= \{f\in {\mathcal A}\mid \exists \alpha\in \mathbb{R}\mid \operatorname {Re} \{e^{i\alpha} (f'(z)- \beta)\}> 0,\;z\in \Delta\}. \] The object of this paper is to establish a sharp relation between \(\beta\) and \((c,\gamma)\) such that \(f\in {\mathcal P}_\beta\) implies \(F_c\in {\mathcal S}_\gamma^*\), \(0\leq \gamma\leq \frac 12\). The cases \(c=0\) (Alexander transform) and \(c=1\) (Libera transform) are considered in detail.
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 1415563
zbMATH Keywords: Alexander transform
zbMATH Keywords: Libera transform
reviewed by: Zbigniew Jerzy Jakubowski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1415563