The Arc Distortion in QH Inner <em>ψ</em>-uniform (or Convex) Domains in Real Banach Spaces
Acta Mathematica Sinica, English Series
Acta Mathematica Sinica,
Chinese Series
Adv Search »  
Acta Mathematica Sinica, English Series  2011, Vol. 27 Issue (10): 2039-2050    DOI: 10.1007/s10114-011-8672-3
Articles Current Issue | Next Issue | Archive | Adv Search  |   
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
 Download: PDF (223 KB)   HTML (1 KB)   Export: BibTeX | EndNote (RIS)      Supporting Info
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.  
E-mail this article
Add to my bookshelf
Add to citation manager
E-mail Alert
Articles by authors
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;

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.
URL:      or
[1] Gehring, F. W., Palka, B. P.: Quasiconformally homogeneous domains. J. Analyse Math., 30, 172-199(1976)
[2] Väisälä, J.: Quasihyperbolic geodesics in convex domains. Results Math., 48, 184-195 (2005)
[3] Gehring, F. W., Osgood, B. G.: Uniform domains and the quasi-hyperbolic metric. J. Analyse Math., 36,50-74 (1979)
[4] Martin, G. J.: Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the hyperbolicmetric. Trans. Amer. Math. Soc., 292, 169-191 (1985)
[5] Väisälä, J.: The free quasiworld, quasiconformal and related maps in Banach spaces. Banach Center Publ.,48, 55-118 (1999)
[6] Väisälä, J.: Free quasiconformality in Banach spaces. I. Ann. Acad. Sci. Fenn. Ser. A I Math., 15, 355-379(1990)
[7] Väisälä, J.: Free quasiconformality in Banach spaces. II. Ann. Acad. Sci. Fenn. Ser. A I Math., 16, 255-310(1991)
[8] Väisälä, J.: Uniform domains. Tohoku Math. J., 40, 101-118 (1988)
[9] Väisälä, J.: Relatively and inner uniform domains. Conformal geom. Dyn., 2, 56-88 (1998)
[10] Martio, O.: Definitions of uniform domains. Ann. Acad. Sci. Fenn. Ser. A I Math., 5, 197-205 (1980)
[11] Näkki, R., Väisälä, J.: John disks. Expo. Math., 9, 3-43 (1991)
[12] Vuorinen, M.: Conformal invariants and quasiregular mappings. J. Analyse Math., 45, 69-115 (1985)
[13] Kl′en, R., Sahoo, S. K., Vuorinen, M.: Uniform continuity and ψ-uniform domains. arXiv: 0812.4369v3 [math. MG]
[14] Broch, O. J.: Geometry of John Disks, Ph. D. Thesis, NTNU, 2004
[15] Gehring, F. W.: Uniform domains and the ubiquitous quasidisks. Jahresber. Deutsch. Math. Verein., 89,88-103 (1987)
[16] Huang, M., Ponnusamy, S., Wang, X., et al.: Uniform domains, John domains and quasi-isotropic domains.J. Math. Anal. Appl., 343, 110-126 (2008)
[17] Kim, K., Langmeyer, N.: Harmonic measure and hyperbolic distance in John disks. Math. Scand., 83,283-299 (1998)
[18] Väisälä, J.: Free quasiconformality in Banach spaces. IV. Analysis and Topology, World Sci. Publ., RiverEdge, NJ, 1998, 697-717
[19] Schäffer, J. J.: Inner diameter, perimeter, and girth of spheres. Math. Ann., 173, 59-82 (1967)
No Similar of article
  Copyright 2012 © Editorial Office of Acta Mathematica Sinica