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On Mori's theorem for quasiconformal maps in the $ n$-space

Authors: B. A. Bhayo and M. Vuorinen
Journal: Trans. Amer. Math. Soc. 363 (2011), 5703-5719
MSC (2000): Primary 30C65
Published electronically: June 7, 2011
MathSciNet review: 2817405
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Abstract: R. Fehlmann and M. Vuorinen proved in 1988 that Mori's constant $ M(n,K)$ for $ K$-quasiconformal maps of the unit ball in $ \mathbf{R}^n$ onto itself keeping the origin fixed satisfies $ M(n,K) \to 1$ when $ K\to 1.$ Here we give an alternative proof of this fact, with a quantitative upper bound for the constant in terms of elementary functions. Our proof is based on a refinement of a method due to G.D. Anderson and M. K. Vamanamurthy. We also give an explicit version of the Schwarz lemma for quasiconformal self-maps of the unit disk. Some experimental results are provided to compare the various bounds for the Mori constant when $ n=2.$

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Additional Information

B. A. Bhayo
Affiliation: Department of Mathematics, University of Turku, FI-20014 Turku, Finland

M. Vuorinen
Affiliation: Department of Mathematics, University of Turku, FI-20014 Turku, Finland

Keywords: Quasiconformal mappings, Hölder continuity
Received by editor(s): September 2, 2009
Published electronically: June 7, 2011
Dedicated: In memoriam: M. K. Vamanamurthy, 5 September 1934–6 April 2009
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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