Modified Distance Ratio Metrics via Domain Diameter and their geometric implications

Abstract

Let $D\subsetneq\mathbb{R}^n,~n\ge 2$, be a domain. In this manuscript, a new version of the Vuorinen's distance ratio metric $j_D$ [{\tt J. Analyse Math.} {\bf 45} (1985), 69--115], denoted by $\zeta_D$, and a version of Gehring-Osgood's distance ratio metric $j_D'$ [{\tt J. Analyse Math.} {\bf 36} (1979), 50--74], denoted by $\zeta_D'$, are introduced to better understand how quasihyperbolic geometry interacts with bounded uniform domains in $\mathbb{R}^n$. We show that the metric $m_D$, introduced in [{\tt arXiv:2505.10964v2}], is the inner metric of $\zeta_D$ and explore their relations to several well-known hyperbolic-type metrics. The paper includes ball inclusion properties of these metrics associated with the metric $m_D$ and other hyperbolic-type metrics. The distortion properties of them are also considered under several important classes of mappings. Furthermore, as an application, we demonstrate that uniform domains can be characterized in terms of metrics $\zeta_D$ and $m_D$.