Extensions of the maximum principle for vector-valued analytic and harmonic functions
type: Item
instance of: scholarly article
title: Extensions of the maximum principle for vector-valued analytic and harmonic functions
zbMATH Open document ID: 0834.30023
author: P. N. Dowling
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-03-18T00:00:00Z
review text: The maximum principle in the complex plane is well known: if \(f(z)\) is a complex valued analytic function on a domain \(G\) in the complex plane and if \(f(z)\) attains its maximum value on \(G\), then \(f(z)\) is a constant function. If \(f(z)\) has values in a complex Banach space, the maximum principle holds iff the Banach space is complex strictly convex. The author extends this theorem to show that a complex Banach space \(X\) is strictly convex if all non-constant analytic functions, defined on the open unit disc in the complex plane, with values in \(X\), have a certain mean growth. This result cannot be extended to the quasi-Banach space and it cannot be uniformized.
reviewed by: Vasile Ionescu
Mathematics Subject Classification ID: 30C80
zbMATH DE Number: 765143
zbMATH Keywords: maximum principle
zbMATH Keywords: complex Banach space
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1095
OpenAlex ID: W2040473126
DOI: 10.1006/JMAA.1995.1095
description: scientific article; zbMATH DE number 765143