Region of variability of two subclasses of univalent functions
type: Item
instance of: scholarly article
title: Region of variability of two subclasses of univalent functions
zbMATH Open document ID: 1114.30014
author: Saminathan Ponnusamy
author: A. Vasudevarao
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-06-06T00:00:00Z
review text: Let \(F_1\) denote the class of functions \(f(z)=z+\dots\), \(| z| <1\), satisfying the condition \(\text{Re}(1+zf"(z)/f'(z))<3/2\). Similarly, the class \(F_2\) is defined by the condition \(\text{Re}(1+zf"(z)/f'(z))>-1/2\). For \(| \lambda| \leq1\) and \(| z_0| <1\), define \[ C_1(\lambda)=\{f\in F_1: f"(0)=-\lambda\},\;\;\;C_2(\lambda)=\{f\in F_2: f"(0)=3\lambda\}, \] \[ V_j(z_0,\lambda)=\{\log f'(z_0): f\in C_j(\lambda)\},\;\;j=1,2. \] The authors determine the regions \(C_1(\lambda)\), \(C_2(\lambda)\). Namely, for \(0\leq\lambda\leq1\) and \(0<| z_0| <1\), the boundary \(\partial V_1(z_0,\lambda)\) is the Jordan curve \(\gamma(\theta)\), \(\theta\in[-\pi,\pi]\), given by \[ \gamma(\theta)=\int_0^{z_0}\frac{\delta(e^{i\theta}z,\lambda)}{z\delta(e^{i\theta}z,\lambda)-1}dz \] where \[ \delta(z,\lambda)=\frac{z+\lambda}{1+\overline{\lambda}z}. \] The boundary \(\partial V_2(z_0,\lambda)\) is described in the same way with obvious arithmetical changes in the rational kernel of the integral.
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C75
zbMATH DE Number: 5161777
zbMATH Keywords: starlike functions
zbMATH Keywords: close-to-convex functions
zbMATH Keywords: univalent functions
zbMATH Keywords: variability region
reviewed by: Dmitri V. Prokhorov
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.11.019
OpenAlex ID: W2072387994
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DOI: 10.1016/J.JMAA.2006.11.019
description: scientific article; zbMATH DE number 5161777