Mathematische Zeitschrift

, Volume 250, Issue 4, pp 885–906

Some inequalities for the Poincaré metric of plane domains

Article

Abstract.

In this paper, the Poincaré (or hyperbolic) metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain Open image in new window In particular, another proof of a recent result of Gardiner and Lakic [7] is given with explicit constant. This and some other constants in this paper involve particular values of complete elliptic integrals and related special functions. A concrete estimate for the hyperbolic distance near a boundary point is also given, from which refinements of Littlewood’s theorem are derived.

Mathematics Subject Classification (1991):

30F45 30A10 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Department of MathematicsUniversity of TurkuTurkuFinland

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