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.
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