Distortion theorems for biholomorphic convex mappings on bounded convex circular domains
type: Item
instance of: scholarly article
title: Distortion theorems for biholomorphic convex mappings on bounded convex circular domains
zbMATH Open document ID: 0942.32017
DOI: 10.1142/S0252959999000321
author: Sheng Gong
author: Taishun Liu
published in: Chinese Annals of Mathematics. Series B
publication date: 2000-08-03T00:00:00Z
review text: The main purpose of this paper is to generalize certain distortion theorems for convex mappings by considering bounded convex circular domains instead of the unit ball. Let \(p=p(z)\) be the Minkowski functional of the domain, and let \(F=F(z,v)\) be the infinitesimal form of the Kobayashi (= Carathéodory) metric. The auhtors give the sharp upper and lower bounds for \(p(J_f(z)v)\) in terms of \(p(z)\) and \(F(z,v)\), where \(J_f\) denotes the jacobian matrix. This generalizes the corresponding estimate on the ball proved by \textit{S. Gong, S. Wang} and \textit{Q. Yu} [Pac. J. Math. 161, No. 2, 287-306 (1993; Zbl 0788.32017)]. In addition, Gong and Liu derive in terms of \(p\), upper and lower bounds for the eigenvalues of the jacobian. The question remains open as to the sharp upper and lower bounds for \(|\det J_f|\) for convex mappings, even on the ball.
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 1385314
zbMATH Keywords: distortion theorem
zbMATH Keywords: convex mapping
zbMATH Keywords: bounded convex circular domain
zbMATH Keywords: Kobayashi metric
zbMATH Keywords: Minkowski functional
reviewed by: Martin Chuaqui
MaRDI profile type: Publication
full work available at URL: s0252959999000321
OpenAlex ID: W1968132313
description: scientific article; zbMATH DE number 1385314