Quasihyperbolic metric and Möbius transformations
HTML articles powered by AMS MathViewer
- by Riku Klén, Matti Vuorinen and Xiaohui Zhang PDF
- Proc. Amer. Math. Soc. 142 (2014), 311-322 Request permission
Abstract:
An improved version of the quasiinvariance property of the quasi- hyperbolic metric under Möbius transformations of the unit ball in ${\mathbb R}^n, n \ge 2,$ is given, and a quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is proved. Finally, several inequalities between the quasihyperbolic metric and other commonly used metrics such as the hyperbolic metric of the unit ball and the chordal metric are established.References
- Lars V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981. MR 725161
- G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Dimension-free quasiconformal distortion in $n$-space, Trans. Amer. Math. Soc. 297 (1986), no. 2, 687–706. MR 854093, DOI 10.1090/S0002-9947-1986-0854093-5
- Glen D. Anderson, Mavina K. Vamanamurthy, and Matti K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. With 1 IBM-PC floppy disk (3.5 inch; HD); A Wiley-Interscience Publication. MR 1462077
- G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Special functions of quasiconformal theory, Exposition. Math. 7 (1989), no. 2, 97–136. MR 1001253
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195
- A. F. Beardon and D. Minda, The diameter of a hyperbolic disc, Complex Anal. Oper. Theory 5 (2011), no. 3, 881–887. MR 2836330, DOI 10.1007/s11785-010-0091-4
- B. A. Bhayo and M. Vuorinen, On Mori’s theorem for quasiconformal maps in the $n$-space, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5703–5719. MR 2817405, DOI 10.1090/S0002-9947-2011-05281-5
- F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Analyse Math. 36 (1979), 50–74 (1980). MR 581801, DOI 10.1007/BF02798768
- F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199. MR 437753, DOI 10.1007/BF02786713
- Peter Hästö, Zair Ibragimov, David Minda, Saminathan Ponnusamy, and Swadesh Sahoo, Isometries of some hyperbolic-type path metrics, and the hyperbolic medial axis, In the tradition of Ahlfors-Bers. IV, Contemp. Math., vol. 432, Amer. Math. Soc., Providence, RI, 2007, pp. 63–74. MR 2342807, DOI 10.1090/conm/432/08300
- Riku Klén, On hyperbolic type metrics, Ann. Acad. Sci. Fenn. Math. Diss. 152 (2009), 49. Dissertation, University of Turku, Turku, 2009. MR 2814337
- Gaven J. Martin and Brad G. Osgood, The quasihyperbolic metric and associated estimates on the hyperbolic metric, J. Analyse Math. 47 (1986), 37–53. MR 874043, DOI 10.1007/BF02792531
- O. Martio, S. Rickman, and J. Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 465 (1970), 13. MR 0267093
- Yu. G. Reshetnyak, Stability theorems in geometry and analysis, Mathematics and its Applications, vol. 304, Kluwer Academic Publishers Group, Dordrecht, 1994. Translated from the 1982 Russian original by N. S. Dairbekov and V. N. Dyatlov, and revised by the author; Translation edited and with a foreword by S. S. Kutateladze. MR 1326375, DOI 10.1007/978-94-015-8360-2
- Pasi Seittenranta, Möbius-invariant metrics, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 511–533. MR 1656825, DOI 10.1017/S0305004198002904
- Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009
- Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174, DOI 10.1007/BFb0077904
- Matti Vuorinen, Quadruples and spatial quasiconformal mappings, Math. Z. 205 (1990), no. 4, 617–628. MR 1082879, DOI 10.1007/BF02571267
- Matti Vuorinen, Metrics and quasiregular mappings, Quasiconformal mappings and their applications, Narosa, New Delhi, 2007, pp. 291–325. MR 2492508
Additional Information
- Riku Klén
- Affiliation: Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland
- Email: ripekl@utu.fi
- Matti Vuorinen
- Affiliation: Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland
- MR Author ID: 179630
- Email: vuorinen@utu.fi
- Xiaohui Zhang
- Affiliation: Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland
- Email: xiazha@utu.fi
- Received by editor(s): August 23, 2011
- Received by editor(s) in revised form: November 16, 2011, and March 13, 2012
- Published electronically: October 4, 2013
- Communicated by: Tatiana Toro
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 311-322
- MSC (2010): Primary 30C65, 51M10
- DOI: https://doi.org/10.1090/S0002-9939-2013-11765-X
- MathSciNet review: 3119205