A distortion theorem for conformal mappings with an application to subharmonic functions
type: Item
instance of: scholarly article
title: A distortion theorem for conformal mappings with an application to subharmonic functions
zbMATH Open document ID: 0714.30026
author: Makoto Masumoto
published in: Hiroshima Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: Let D be a finitely connected domain in the complex plane such that every component of \({\mathbb{C}}\)-D contains at least two points. If there exist \(\rho >0\) and \(\theta\in (0,1)\) such that for every \(\omega\in \partial D\) a closed circular sector with vertex at \(\omega\) which is congruent to \(S(\rho,\theta)=\{w\in {\mathbb{C}}:\) \(0\leq | w| \leq \rho\), \(0\leq \arg w\leq \pi \theta \}\) lies in \({\mathbb{C}}\)-D, then D is said to satisfy an exterior \(\theta\)-wedge condition. Suppose that \(D_ 0\) is a finitely connected domain whose boundary components are simple analytic curves. The author first proves that if f is a conformal mapping of \(D_ 0\) onto D, and if \(\lambda_{D_ 0}(z)\) is the Poincaré metric for \(D_ 0\), then there exists \(m>0\) such that for all \(z\in D_ 0\), \[ \frac{1}{| f'(z)|}\leq m\lambda_{D_ 0}(z)^{1-\theta}. \] This distortion theorem is then applied to prove the following, which generalizes a theorem of Suzuki. Suppose that D satisfies an exterior \(\theta\)-wedge condition. For \(0<p<\infty\), set \(\beta (p,\vartheta)=2- \min \{1,p\}/(2-\theta)\), and let \(\delta_ D(z)\) be the distance from z in D to \(\partial D\). If a non-negative subharmonic function s on D satisfies \[ \iint_{D}\delta_ D(z)^{-\beta (p,\theta)}s(z)^ p dx dy<\infty, \] then \(s\equiv 0\) on D.
Mathematics Subject Classification ID: 30C80
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 4175339
zbMATH Keywords: wedge condition
zbMATH Keywords: Poincaré metric
reviewed by: F. D. Lesley
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4175339