Convex mappings on the unit ball of \(\mathbb C^ n\)
type: Item
instance of: scholarly article
title: Convex mappings on the unit ball of \(\mathbb C^ n\)
zbMATH Open document ID: 0846.32006
DOI: 10.1007/BF02788776
author: Kevin A. Roper
author: Ted J. Suffridge
published in: Journal d'Analyse Mathématique
publication date: 1996-09-29T00:00:00Z
review text: The authors study convex mappings of the unit ball into \(\mathbb C^n\), i.e. mappings \(F : B \to \mathbb C^n\) such that \(F\) is univalent, \(F(0) = 0\), \(dF(0) = I\), and \(F(B)\) is convex. The main result of the paper is the following theorem: Theorem 1. Suppose \(f : \Delta \to \mathbb C\) is a univalent convex function on the unit disk \(\Delta \subset \mathbb C\) with \(f(0) = 0\), \(f'(0) = 1\). Form a new function \(F : B \to \mathbb C^2\), where \(B\) is the unit ball in \(\mathbb C^2\) and \(F(2,w) = (f(z), w \sqrt {f'(z)})\). Then \(f\) is a convex mapping. Theorem 1 can be generalized to any complex dimension.
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 807411
zbMATH Keywords: univalent convex function
zbMATH Keywords: unit disk
zbMATH Keywords: unit ball
reviewed by: A. V. Isaev
MaRDI profile type: Publication
cites work: Φ-Like Holomorphic Functions In C n and Banach Spaces
cites work: A property of convex conformal maps
cites work: Q5815016
cites work: The principle of subordination applied to functions of several variables
cites work: Q4129080
full work available at URL: https://doi.org/10.1007/bf02788776
OpenAlex ID: W2026173772
description: scientific article; zbMATH DE number 807411