Comparison and distortion properties of new distance ratio metrics

Abstract

In 1979, Gehring and Osgood introduced the distance ratio metric, and later Vuorinen proposed another version of this metric. This manuscript investigates a new variant of these metrics defined on bounded domains in $R^n$, $n\ge 2$. We show that the metric $m_D$, recently introduced by the authors, is the inner metric of one of the newly defined metrics corresponding to Vuorinen's version of the distance ratio metric. We also explore its relationships with several well-known hyperbolic-type metrics. The paper presents ball inclusion properties of the metrics associated with $m_D$ and other hyperbolic-type metrics. Their distortion properties are also examined under several important classes of mappings. Furthermore, as an application, we demonstrate that these metrics can be used to characterize uniform domains.