Modulus of continuity of harmonic functions
type: Item
instance of: scholarly article
title: Modulus of continuity of harmonic functions
zbMATH Open document ID: 0672.31001
DOI: 10.1007/BF02791117
author: S. H. Smith
published in: Journal d'Analyse Mathématique
publication date: 1988-01-01T00:00:00Z
review text: Suppose that u(z) is a harmonic function in a plane domain G and that the modulus of continuity of u(z) is majorised by a nondecreasing function \(\mu\) (t), \(\mu\) (2t)\(\leq 2\mu (t)\), on the boundary \(\partial G\). What kind of upper bound can be obtained for \(| u(z_ 1)-u(z_ 2)|\) when \(z_ 1,z_ 2\in \bar G?\) Making use of various estimates of the harmonic measure the author gives quite explicit bounds such as \[ | u(z_ 1)-u(z_ 2)| \leq \mu (| z_ 1-z_ 2|)(K_ 1+K_ 2S(| z_ 1-z_ 2|)), \] for \(z_ 1,z_ 2\in \bar G\), where \(K_ 1\) and \(K_ 2\) are some constants (at least one of them must depend on G) and S(t) is some function.
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 4099646
zbMATH Keywords: harmonic
zbMATH Keywords: modulus of continuity
zbMATH Keywords: upper bound
zbMATH Keywords: harmonic measure
reviewed by: M. I. Sodin
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02791117
OpenAlex ID: W2044190223
description: scientific article; zbMATH DE number 4099646