A GENERALIZATION OF VUORINEN'S DISTANCE RATIO METRIC IN METRIC SPACES AND BI-LIPSCHITZ EQUIVALENT HYPERBOLIC-TYPE METRICS

Abstract

We prove in the setting of a general metric space (X, d) the bi-Lipschitz equivalence of generalized versions of Vuorinen's distance ratio metric, Gehring-Osgood metric, Dovgoshey-Hariri-Vuorinen metric, Nikolov-Andreev metric and Ibragimov metric. For the generalized Vuorinen's distance ratio metric j on the complement of a nonempty closed subset M of X we show that the identity map of X\M between (X\M,d) and (X\M,j) is 1-quasiconformal. We also provide su_cient conditions for the completeness of (X\M,j), that is equivalent to the completeness of X\M with each of the above mentioned metrics.

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