Remarks on the scale invariant Cassinian metric

TL;DR

This paper investigates the properties of the scale invariant Cassinian metric, establishing sharp inequalities relating it to the hyperbolic metric and analyzing its distortion under M"obius transformations in specific domains.

Contribution

It provides new sharp comparison and distortion inequalities for the scale invariant Cassinian metric in the unit ball and upper half space.

Findings

Sharp comparison inequalities between Cassinian and hyperbolic metrics.

Sharp distortion inequalities under M"obius transformations.

Results specific to the unit ball and upper half space domains.

Abstract

We study the geometry of the scale invariant Cassinian metric and prove sharp comparison inequalities between this metric and the hyperbolic metric in the case when the domain is either the unit ball or the upper half space. We also prove sharp distortion inequalities for the scale invariant Cassinian metric under M\"obius transformations.