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FREELY QUASICONFORMAL MAPS AND DISTANCE RATIO METRIC

Part of: Riemann surfaces Geometric function theory

Published online by Cambridge University Press:  09 September 2014

YAXIANG LI ,
SAMINATHAN PONNUSAMY  and
MATTI VUORINEN
YAXIANG LI
Affiliation:
College of Science, Central South University of Forestry and Technology, Changsha, Hunan 410004, PR China email yaxiangli@163.com
SAMINATHAN PONNUSAMY*
Affiliation:
Indian Statistical Institute (ISI) Chennai Centre, SETS (Society for Electronic Transactions and Security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India email samy@isichennai.res.in, samy@iitm.ac.in
MATTI VUORINEN
Affiliation:
Department of Mathematics and Statistics, University of Turku, FIN-20014 Turku, Finland email vuorinen@utu.fi
*
✉samy@isichennai.res.in, samy@iitm.ac.in
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Abstract

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Suppose that $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ and $E'$ denote real Banach spaces with dimension at least 2 and that $D\subset E$ and $D'\subset E'$ are domains. Let $\varphi :[0,\infty )\to [0,\infty )$ be a homeomorphism with $\varphi (t)\geq t$. We say that a homeomorphism $f: D\to D'$ is $\varphi $-FQC if for every subdomain $D_1 \subset D$, we have $\varphi ^{-1} (k_D(x,y))\leq k_{D'} (f(x),f(y))\leq \varphi (k_D(x,y))$ holds for all $x,y\in D_1$. In this paper, we establish, in terms of the $j_D$ metric, a necessary and sufficient condition for a homeomorphism $f: E \to E'$ to be FQC. Moreover, we give, in terms of the $j_D$ metric, a sufficient condition for a homeomorphism $f: D\to D'$ to be FQC. On the other hand, we show that this condition is not necessary.

Keywords

quasihyperbolic metricquasiconformal mappingFQC mappingdistance ratio metric

MSC classification

Secondary: 30C65: Quasiconformal mappings in $^n$, other generalizations 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions) 30C20: Conformal mappings of special domains
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 3 , December 2014 , pp. 383 - 390
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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