Loewner chains and parametric representation in several complex variables
type: Item
instance of: scholarly article
title: Loewner chains and parametric representation in several complex variables
zbMATH Open document ID: 1029.32004
DOI: 10.1016/S0022-247X(03)00127-6
author: I. Graham
author: G. Kohr
author: Mirela Kohr
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(B\) be the unit ball of \(\mathbb{C}^n\) with respect to an arbitrary norm. The authors study certain properties of Loewner chains and their transition mappings on the unit ball \(B\). They show that any Loewner chain \(f(z,t)\) and the transition mapping \(v(z,s,t)\) associated to \(f(z,t)\) satisfy locally Lipschitz conditions in \(t\) locally uniformly with respect to \(z\in B\). Moreover, they prove that a mapping \(f\in H(B)\) has parametric representation if and only if there exists a Loewner chain \(f(z,t)\) such that the family \(\{e^{-t}f(z,t)\}_{t\geq 0}\) is a normal family on \(B\) and \(f(z)=f(z,0)\). The authors also show that univalent solutions \(f(z,t)\) of the generalized Loewner differential equation in higher dimensions are unique when \(\{e^{-t}f(z,t)\}_{t\geq 0}\) is a normal family in \(B\). Finally, they show that the set of mappings which have parametric representation on \(B\) is compact.
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 32A17
Mathematics Subject Classification ID: 32A19
zbMATH DE Number: 1956872
zbMATH Keywords: Loewner chain
zbMATH Keywords: transition mapping
zbMATH Keywords: parametric representation
zbMATH Keywords: univalent mapping
zbMATH Keywords: Loewner differential equation
reviewed by: Martin Chuaqui
MaRDI profile type: Publication
cites work: Über die Lösungsstruktur einer Differentialgleichung in der konformen Abbildung.
cites work: Q3952359
cites work: Q3699128
cites work: Applications of subordination chains to starlike mappings in \(\mathbb{C}^ n\)
cites work: Parametric Representation of Univalent Mappings in Several Complex Variables
cites work: Extension operators for locally univalent mappings
cites work: An Extension Theorem and Subclasses of Univalent Mappings in Several Complex Variables
cites work: Q4444884
cites work: Loewner chains and the Roper-Suffridge extension operator
cites work: Starlike mappings on bounded balanced domains with \(C^1\)-plurisubharmonic defining functions
cites work: Q4438926
cites work: Subordination chains and univalence of holomorphic mappings on bounded balanced pseudoconvex domains
cites work: The growth theorem and quasiconformal extension of strongly spirallike mappings of type α
cites work: Loewner chains and quasiconformal extension of holomorphic mappings
cites work: Q4447166
cites work: ON THE PARAMETRIC REPRESENTATION OF STARLIKE MAPS OF THE UNIT BALL IN CN INTO CN
cites work: Q5628476
cites work: Subordination chains and univalence of holomorphic mappings in C\(^n\)
cites work: Subordination chains and quasiconformal extension of holomorphic maps in C^n
cites work: Close-to-starlike holomorphic functions of several variables
cites work: Q4052427
cites work: Q3475550
cites work: Q3469741
cites work: Q4870362
cites work: Q3996430
cites work: Convexity properties of holomorphic mappings in ℂⁿ
cites work: Starlike and convex maps in Banach spaces
full work available at URL: s0022-247x(03)00127-6
OpenAlex ID: W1986016332
description: scientific article; zbMATH DE number 1956872