Explicit relations for the reduced module and harmonic measure
type: Item
instance of: scholarly article
title: Explicit relations for the reduced module and harmonic measure
zbMATH Open document ID: 0819.30015
DOI: 10.1007/BF01062956
author: V. I. Belyj
author: E. V. Kravchuk
published in: Ukrainian Mathematical Journal
publication date: 1994-09-26T00:00:00Z
review text: Using conformal mappings of simply connected domains onto the unit disk or its exterior, the values of some reduced moduli of families of curves separating connected sets are calculated in an elementary way. These reduced moduli are introduced to formulate the theorems, but the specific methods of ``extremal lengths'' don't seem (from this short communication) to have been applied here. -- In the particular case of the harmonic measure \(\omega (z, \alpha)\) of a boundary arc \(\alpha\) at an interior point \(z\) in a simply connected domain \(G\), the curves considered here are those separating \(z\) from \(\alpha\), including the closed curves surrounding \(z\). The elementary relation obtained in this case between \(\omega\) and the reduced modulus \(\mu\) can be interpreted (and proved directly) as the relation between \(\omega\) and the (conformal invariant) ratio of two ``conformal radii'' \(R_ z\) and \(\widetilde R_ z\), namely \(\sin^ 2 (\pi \omega/2) = \widetilde R_ z/R_ z\). The authors compare their elementary relation between \(\omega\) and \(\mu\) with the known relation (in terms of elliptic integrals) between \(\omega\) and the extremal length of the sub-family of curves with end-points on \(\partial G \backslash \alpha\), separating \(z\) from \(\alpha\). For this last relation they quote a paper of the reviewer (1955); but this idea stays already in the Oklahoma College lecture notes ``Conformal mapping'' of L. V. Ahlfors (Summer 1951) (page 91), of which also the reviewer had unfortunately no knowledge at that time.
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 30C70
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 635754
zbMATH Keywords: extremal lengths
zbMATH Keywords: disk
zbMATH Keywords: reduced moduli
zbMATH Keywords: harmonic measure
reviewed by: Joseph Hersch
MaRDI profile type: Publication
cites work: Q3937799
cites work: Q5527180
cites work: Q5772332
cites work: Confromal invariants and function-theoretic null-sets
cites work: Longueurs extrémales et théorie des fonctions
full work available at URL: https://doi.org/10.1007/bf01062956
OpenAlex ID: W2055026489
description: scientific article; zbMATH DE number 635754