A note on \(\phi\)-uniform domains
type: Item
instance of: scholarly article
title: A note on \(\phi\)-uniform domains
zbMATH Open document ID: 1377.30036
author: Debanjan Nandi
published in: The Journal of Analysis
publication date: 2017-05-03T00:00:00Z
review text: Let \(\Omega\) be a domain in \(\mathbb R^n\) and \(d(x)\) be the euclidean distance of a point in \(\Omega\) to its boundary. A domain \(\Omega\) is uniform iff the quasihyperbolic distance \(k(x,y)\) satisfies the inequality \(k(x,y) \leq C \log \left(1+\frac{|x-y|}{d(x)\wedge d(y)} \right)\) for all \(x,y \in \Omega\) (\(C\geq 1\)). Let \(\phi: [0,\infty) \to [0,\infty)\) be a homeomorphism. A domain \(\Omega\) is said to be \(\phi\)-uniform if \(k(x,y) \leq \phi \left( \frac{|x-y|}{d(x)\wedge d(y)} \right)\) for all \(x,y \in \Omega\). The author constructs an example of a quasiconvex, simply connected and bounded planar domain which fails to be \(\phi\)-uniform for any homeomorphism.
Mathematics Subject Classification ID: 30F45
zbMATH DE Number: 6710824
zbMATH Keywords: quasihyperbolic metric
zbMATH Keywords: \(\phi\)-uniform domains
Wikidata QID: Q110235293
reviewed by: Vladimir Mityushev
MaRDI profile type: Publication
full work available at URL: s41478-016-0004-7
OpenAlex ID: W2513511082
cites work: Uniform domains and the quasi-hyperbolic metric
cites work: POSITIVE HARMONIC FUNCTIONS IN UNIFORM AND ADMISSIBLE DOMAINS
cites work: On the stability of \(\varphi\)-uniform domains
cites work: John disks
cites work: Conformal invariants and quasiregular mappings
DOI: 10.1007/S41478-016-0004-7
description: scientific article; zbMATH DE number 6710824