On Uniformity Exponents of $$\varphi $$ -Uniform Domains

Let \(G\subsetneq {\mathbb {R}}^n\) be a domain, where \(n\ge 2\). Let \(k_G\) and \(j_G\) be the quasihyperbolic metric and the distance ratio metric on G, respectively. In the present paper, we prove that the identity map of \((G,k_G)\) onto \((G,j_G)\) is quasisymmetric if and only if it is bilipschitz. To classify domains of \({\mathbb {R}}^n\) into various types according to the behaviors of their quasihyperbolic metrics, we define a uniformity exponent for every proper subdomain of \({\mathbb {R}}^n\) and prove that this exponent may assume any value in \(\{0\}\cup [1,\infty ]\). Moreover, we study the properties of domains of uniformity exponent 1 and show by an example that such a domain may be neither quasiconvex nor accessible.