Some results on rigidity of holomorphic mappings
type: Item
instance of: scholarly article
title: Some results on rigidity of holomorphic mappings
zbMATH Open document ID: 0817.32012
author: Yoshihisa Kubota
published in: Kodai Mathematical Journal
publication date: 1994-12-19T00:00:00Z
review text: The rigidity property of holomorphic mappings in complex normed spaces is investigated in this paper. For two such spaces \(X\) and \(Y\), let \(D_ 1\) be a balanced domain in \(X\), \(D_ 2\) be a bounded convex domain in \(Y\), and \(f : D_ 1 \to D_ 2\) a holomorphic mapping. Three theorems are proved: The first is a generalization of the Schwarz lemma, that gives an upper bound for \(\mu_{D_ 2} (f(x))\), \(x \in D_ 1\), where \(\mu_{D_ 2}\) is the Minkowski functional of \(D_ 2\). As its corollary, a result on the extremal mappings is deduced, i.e., if \(f\) satisfies \(\| f(x) \| = \| x \|\) for all \(x \in X\), then \(f\) is linear. The second theorem concerns the lower bound for \(\mu_{D_ 2} (f(x))\). At last, the pointwise limit of a sequence of automorphisms of a bounded domain is considered; with the above two theorems it is proved that the limit mapping is one-to-one.
reviewed by: Na Jisheng
Mathematics Subject Classification ID: 32H02
zbMATH DE Number: 701567
zbMATH Keywords: complex normed space
zbMATH Keywords: rigidity
zbMATH Keywords: balanced domain
zbMATH Keywords: holomorphic mapping
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039963
OpenAlex ID: W2077565612
cites work: Q3683019
cites work: Q3892778
cites work: Automorphism Groups of Bounded Domains in Banach Spaces
cites work: SCHWARZ'S LEMMA IN NORMED LINEAR SPACES
cites work: A continuous form of Schwarz's lemma in normed linear spaces
cites work: Q5558683
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cites work: Q3958740
DOI: 10.2996/KMJ/1138039963
description: scientific article; zbMATH DE number 701567