Koebe invariant functions and extremal problems for holomorphic mappings in the unit ball of \(\mathbb C^n\)
type: Item
instance of: scholarly article
title: Koebe invariant functions and extremal problems for holomorphic mappings in the unit ball of \(\mathbb C^n\)
zbMATH Open document ID: 1143.32009
DOI: 10.1007/BF03321652
author: John A. Pfaltzgraff
author: Ted J. Suffridge
published in: Computational Methods and Function Theory
publication date: 2008-05-30T00:00:00Z
full work available at URL: cmf07026.htm
review text: Let \(B\) be the Euclidean ball in \(\mathbb{C}^{n},n\geq2\). Consider \(\sigma\) a holomorphic automorphism of the ball \(B\). A holomorphic mapping \(g:B\rightarrow\mathbb{C}^{n}\) that is invertible in a neighborhood of \(g(\sigma(0))\) is said to be invariant if \(\Lambda_{\sigma}(g)=g\), where \[ \Lambda_{\sigma}(g)(z)=D\sigma(0)^{-1}Dg(\sigma(0))^{-1} \left(g(\sigma(z))-g(\sigma(0))\right). \] In the paper, the authors give a complete characterization of all invariant functions and show how to construct invariant functions for arbitrary \(n\geq2\). A connection between the invariance concept and the solution of extremal problems in certain linear invariant families is also given.
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C55
zbMATH DE Number: 5281676
zbMATH Keywords: locally biholomorphic mapping
zbMATH Keywords: linear invariance
zbMATH Keywords: invariant function
zbMATH Keywords: extremal function
reviewed by: Dorina Raducanu
MaRDI profile type: Publication
cites work: Q4659280
cites work: Q4444884
cites work: Univalent mappings associated with the Roper-Suffridge extension operator
cites work: A modification of the Roper-Suffridge extension operator
cites work: Extreme points for convex mappings of \(B_n \)
cites work: An extension theorem and linear invariant families generated by starlike maps
cites work: Norm order and geometric properties of holomorphic mappings in \(\mathbb{C}^n\)
cites work: Linear-invariante Familien analytischer Funktionen. I
cites work: Linear-invariante Familien analytischer Funktionen. II
cites work: Convex mappings on the unit ball of \(\mathbb C^ n\)
OpenAlex ID: W2056149623
description: scientific article; zbMATH DE number 5281676