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The Journal of Analysis

, Volume 24, Issue 1, pp 57–66 | Cite as

Geometric properties of \(\varphi \)-uniform domains

  • Peter Hästö
  • Riku KlénEmail author
  • Swadesh Kumar Sahoo
  • Matti Vuorinen
Original Research Paper

Abstract

We consider proper subdomains G of \({\mathbb R}^n\) and their images \(G'=f(G)\) under quasiconformal mappings f of \({\mathbb R}^n\). We compare the distance ratio metrics of G and \(G'\); as an application we show that \(\varphi \)-uniform domains are preserved under quasiconformal mappings of \({\mathbb R}^n\). A sufficient condition for \(\varphi \)-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If \(G\subset {\mathbb R}^n\) is \(\varphi \)-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar \(\varphi \)-uniform domain whose complement is not \(\psi \)-uniform for any \(\psi \).

Keywords

The distance ratio metric The quasihyperbolic metric Uniform and \(\varphi \)-uniform domains John domains Quasiconformal and quasisymmetric mappings 

Mathematics Subject Classification

Primary 30F45; Secondary 30C65 30L05 30L10 

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Copyright information

© Forum D'Analystes, Chennai 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of OuluOuluFinland
  2. 2.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  3. 3.Discipline of MathematicsIndian Institute of Technology IndoreIndoreIndia

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