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1. Introduction The Apollonian metric αD isa generalization of the hyperbolic metric which is defined in arbitrary open sets D of R n. It was introduced by Beardon in 1 998 [3], but it later turned out that the same metric had been studied previously by Barbilian [1](see [1 9], and for some further developments,[6, 7]). A very popular generalization of the hyperbolic metric in higher dimensions is the quasihyperbolic metric [1 0