The Arc Distortion in QH Inner <em>ψ</em>-uniform (or Convex) Domains in Real Banach Spaces
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Acta Mathematica Sinica, English Series  2011, Vol. 27 Issue (10): 2039-2050    DOI: 10.1007/s10114-011-8672-3
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The Arc Distortion in QH Inner ψ-uniform (or Convex) Domains in Real Banach Spaces
Man Zi HUANG, Xian Tao WANG
Department of Mathematics, Hu'nan Normal University, Changsha 410081, P. R. China
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Abstract Let D and D′ be domains in real Banach spaces of dimension at least 2. The main aim of this paper is to study certain arc distortion properties in the quasihyperbolic metric defined in real Banach spaces. In particular, when D′ is a QH inner ψ-uniform domain with ψ being a slow (or a convex domain), we investigate the following: For positive constants c, h,C,M, suppose a homeomorphism f : DD′ takes each of the 10-neargeodesics in D to (c, h)-solid in D′. Then f is C-coarsely M-Lipschitz in the quasihyperbolic metric. These are generalizations of the corresponding result obtained recently by Väisälä.  
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Key wordsUniform domain   QH ψ-uniform domain   inner uniform domain   QH inner ψ-uniform domain   convex domain   quasihyperbolic geodesic   neargeodesic   quasiconvexity   real Banach space     
Received: 2008-12-30;
Fund:

Supported by National Natural Science Foundation of China (Grant No. 11071063), Tianyuan Foundation (Grant No. 10926068) and Scientific Research Fund of Hunan Provincial Education Department (Grant No. 09C635)

Cite this article:   
. The Arc Distortion in QH Inner ψ-uniform (or Convex) Domains in Real Banach Spaces[J]. Acta Mathematica Sinica, English Series, 2011, 27(10): 2039-2050.
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http://www.actamath.com/Jwk_sxxb_en//EN/10.1007/s10114-011-8672-3      or     http://www.actamath.com/Jwk_sxxb_en//EN/Y2011/V27/I10/2039
 
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