Abstract.
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.
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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.
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Sugawa, T., Vuorinen, M. Some inequalities for the Poincaré metric of plane domains. Math. Z. 250, 885–906 (2005). https://doi.org/10.1007/s00209-005-0782-0
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DOI: https://doi.org/10.1007/s00209-005-0782-0