, Volume 250, Issue 4, pp 885-906
Date: 15 Apr 2005

Some inequalities for the Poincaré metric of plane domains

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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.

This research was carried out during the first-named author’s visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS.