Quasi-convex mappings of order \(\alpha\) on the unit polydisk in \(\mathbb{C}^n\)
type: Item
instance of: scholarly article
title: Quasi-convex mappings of order \(\alpha\) on the unit polydisk in \(\mathbb{C}^n\)
zbMATH Open document ID: 1209.32006
DOI: 10.1216/RMJ-2010-40-5-1619
author: Xiaosong Liu
author: Ming-Sheng Liu
published in: Rocky Mountain Journal of Mathematics
publication date: 2010-12-21T00:00:00Z
review text: Let \(X\) be a complex Banach space with the norm \(\left\|.\right\|\), let \(X^{*}\) denotes the dual space of \(X\), \(E\) be the open unit ball in \(X\). Also, let \(U^{n}\) be the open unit polydisc in \(\mathbb{C}^{n}\). For each \(x\in X\setminus\left\{0\right\}\) consider \(T(x)=\left\{T_{*}\in X^{*}:\left\|T_{x}\right\|=1, T_{x}(x)=\left\|x\right\|\right\}\). Let \(D^{n}f\) denotes the nth Fr\(\acute{e}\)chet derivative of a holomorphic mapping \(f:E\rightarrow X\). Suppose \(\alpha\in[0,1)\). A normalized locally biholomorphic mapping \(f:E\rightarrow X\) is said to be quasi-convex of type \(B\) and order \(\alpha\) if \[ Re\left\{T_{x}\left[(Df(x))^{-1}(D^{2}f(x)(x^{2})+Df(x)x)\right]\right\}\geq\alpha\left\|x\right\|,\quad x\in E. \] In the paper the authors obtain a sufficient condition for the quasi-convex mappings of type \(B\) and order \(\alpha\) on \(E\). Sharp estimations of all homogeneous expansions are established for quasi-convex mappings of type \(B\) and order \(\alpha\) defined on \(U^{n}\), which satisfy some conditions.
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 5828485
zbMATH Keywords: \(k\)-fold symmetric
zbMATH Keywords: convex function of order \(\alpha\)
zbMATH Keywords: quasiconvex mapping of type B and order \(\alpha\)
zbMATH Keywords: estimation of homogeneous expansion
reviewed by: Dorina Raducanu
MaRDI profile type: Publication
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full work available at URL: rmj-2010-40-5-1619
OpenAlex ID: W2093997422
description: scientific article; zbMATH DE number 5828485