Some inequalities for the Poincaré metric of plane domains

: Sugawa T., Vuorinen M.

: 2005

: Mathematische Zeitschrift

: 250

: 4

: 885

: 906

: 22

: 0025-5874

DOI: https://doi.org/10.1007/s00209-005-0782-0

: http://api.elsevier.com/content/abstract/scopus_id:23744456000

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 [InlineMediaObject not available: see fulltext.] In particular another proof of a recent result of Gardiner and Lakic 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. © Springer-Verlag Berlin Heidelberg 2005.