Quasisymmetry and Quasihyperbolicity of Mappings on John Domains

Suppose that G is a proper subdomain of R-n, f : G -> Y is a homeomorphism with a continuous extension to the inner boundary of G, i.e., the boundary of G with respect to the corresponding inner metric, where (Y, d') stands for a locally compact, noncomplete and rectifiably connected metric space, and that G' = f (G) is uniform in Y. The purpose of this paper is to prove that G is a John domain if f is M-quasihyperbolic in G and the restriction of f on the inner boundary of G is eta-quasisynunetric with respect to the inner metric.