SUGAWA, Toshiyuki and VUORINEN, Matti

*Some inequalities for the Poincar'e metric of plane domains*

### Abstract

In this paper, the Poincar'e (or hyperbolic) metric and the associated
distance are investigated for a plane domain based on the detailed properties of
those for the particular domain **C**-{0,1}. 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.

submission: 23 April, 2002

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