Mathematische Zeitschrift

, Volume 250, Issue 4, pp 885–906

Some inequalities for the Poincaré metric of plane domains


DOI: 10.1007/s00209-005-0782-0

Cite this article as:
Sugawa, T. & Vuorinen, M. Math. Z. (2005) 250: 885. doi:10.1007/s00209-005-0782-0


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 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):


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