The Apollonian inner metric and uniform domains

Abstract

Key words Apollonian metric, inner metric, quasihyperbolic metric, uniform domain, A-uniform domain, Apollonian quasiconvex domain and quasi-isotropic domain MSC (2000) Primary: 30F45; Secondary: 30C65In this paper we characterize uniform domains in terms of the Apollonian inner metric and the j-metric, thus providing solutions to two open problems given in [16]. We also discuss the relationships among uniform, A-uniform, Apollonian quasiconvex and quasi-isotropic domains.

Inner Uniform Domains and the Apollonian Inner Metric

“…As a matter of fact, the above inequality appeared in [6] in a form with an additive constant on the right hand side: it was shown by Vuorinen [25, 2.50] that the additive constant can be chosen to be 0. Moreover, in [13], the authors proved the following. See [3,4,6,13,15,16,17,21,22,24] for more details on uniform domains and inner uniform domains.…”

Uniform domains, John domains and quasi-isotropic domains

“…Also there exist doubly connected domains which are John domains, but not quasi-isotropic. For instance, [14,Example 3.11] gives that B \ {0} is not quasi-isotropic but is clearly a John domain. The paper is organized as follows: Section 2 contains preliminary information including basic definitions, notational descriptions, and elementary results for proving our main results.…”

Inclusion relations of hyperbolic type metric balls II

“…The quasihyperbolic metric has recently been studied in [2,11]. There are also other hyperbolic type metrics like the distance ratio metric and the Apollonian metric, which have lately been studied by various authors [3,4].…”