Universally prestarlike functions as convolution multipliers
type: Item
instance of: scholarly article
title: Universally prestarlike functions as convolution multipliers
zbMATH Open document ID: 1181.30011
author: Stephan Ruscheweyh
author: Luis Salinas
published in: Mathematische Zeitschrift
publication date: 2009-10-23T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/140997
review text: Let \({\mathcal H}_{1}(\Omega)\) denote the set of functions \(f\) of the form \(f(z) = z + a_2z^2 + \dots\) that are analytic in a domain \(\Omega\) containing the origin. By \({\mathcal R}_{\alpha}\) we denote the class of functions \(f \in {\mathcal H}_{1}(D)\) which are prestarlike of order \(\alpha\) (with \(\alpha \leq 1\)) in the unit disk \(D\). For \(\gamma \in {\mathbb C} \setminus \{0\}\) and \(\rho \in [0,1]\), define \[ \Omega_{\gamma, \rho} = w_{\gamma, \rho}(D),\;\;\;\;w_{\gamma, \rho}(z) = \frac{\gamma z}{1- \rho z},\;z \in D. \] A function \(f \in {\mathcal H}_{1}(\Omega_{\gamma, \rho})\) is called prestarlike of order \(\alpha\) (\(\alpha \leq 1\)) in \(\Omega_{\gamma, \rho}\) if \[ \frac{1}{\gamma}\,f(w_{\gamma, \rho}(z)) \in {\mathcal R}_{\alpha}. \] Let \(\Lambda\) be the slit domain \(\mathbb C \setminus [1, \infty)\). A function \(f \in {\mathcal H}_{1}(\Lambda)\) is called universally prestarlike of order \(\alpha\) (\(\alpha < 1\)) if \(f\) is prestarlike of order \(\alpha\) in all sets \(\Omega_{\gamma, \rho}\) for which \(1 \notin \Omega_{\gamma, \rho}\). The authors prove in particular that the classes of universally prestarlike functions (with \(\alpha\) fixed) are closed under convolution, and that their members, as Hadamard multipliers, also preserve the prestarlikeness (of the same order) of functions in arbitrary circular domains containing the origin.
reviewed by: Agnieszka Wisniowska-Wajnryb
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 44A60
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 5620115
zbMATH Keywords: universally prestarlike functions
zbMATH Keywords: universally convex functions
zbMATH Keywords: convolution invariance
zbMATH Keywords: Bernardi-Libera transform
MaRDI profile type: Publication
OpenAlex ID: W2081651212
cites work: Q5558293
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cites work: Q4787437
cites work: The Hadamard product and linear transformations of classes of analytic functions
cites work: Q3820868
DOI: 10.1007/S00209-008-0433-3
description: scientific article; zbMATH DE number 5620115