Criterion for the family of rounded \(\varepsilon\) starlike mappings
type: Item
instance of: scholarly article
title: Criterion for the family of rounded \(\varepsilon\) starlike mappings
zbMATH Open document ID: 1019.32009
DOI: 10.1016/S0022-247X(02)00337-2
author: Sheng Gong
author: Taishun Liu
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The authors introduce the family of \(\varepsilon\) starlike mappings and study the criterion for this family. The results obtained for this family cover the known results about the family of convex mappings and the family of starlike mappings, and describe how one family can transit to another one. They give the criterion for the family of \(\varepsilon\) starlike mappings on the unit ball in the complex Banach space and also in a bounded convex circular domain in \(\mathbb{C}^n\). One of the results obtained is: Let \(X,Y\) be two complex Banach spaces, and let \(B\subset X\) be the open unit ball. If \(f:B\to Y\) is a locally holomorphic mapping on \(B\) with \(f(0)=0\), then \(f\) is \(\varepsilon\) starlike if and only if \[ \text{Re} \Bigl\{T_x\biggl[ \bigl(Df(x) \bigr)^{-1}\bigl( f(x)-\varepsilon f(y)\bigr) \biggr]\Bigr\}\geq 0 \] holds for any \(x\in B\) and \(y\in B\) whenever \(\|y\|\leq \|x\|<1\), where \(T_x\in T(x)\), \(Df(x)\) is the Fréchet derivative of \(f\) at \(x\).
reviewed by: Arun Kumar Agarwal
Mathematics Subject Classification ID: 32E05
Mathematics Subject Classification ID: 32H02
zbMATH DE Number: 1866654
zbMATH Keywords: starlike mappings
zbMATH Keywords: convex mappings
zbMATH Keywords: unit ball
zbMATH Keywords: complex Banach space
zbMATH Keywords: bounded convex circular domain
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1866654