Quasihyperbolic metric and M\"obius transformations

TL;DR

This paper improves the understanding of how the quasihyperbolic metric behaves under M"obius and quasiconformal transformations, establishing sharper invariance properties and relations to other metrics.

Contribution

It provides an improved quasiinvariance property of the quasihyperbolic metric under M"obius transformations and establishes sharp local quasiinvariance under quasiconformal mappings.

Findings

Enhanced quasiinvariance bounds for M"obius transformations

Sharp local quasiinvariance under quasiconformal mappings

New inequalities relating quasihyperbolic, hyperbolic, and chordal metrics

Abstract

An improved version of quasiinvariance property of the quasihyperbolic metric under M\"obius transformations of the unit ball in ${\mathbb R}^n, n \ge 2,$ is given. Next, a quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is proved. Finally, several inequalities between the quasihyperbolic metric and other commonly used metrics such as the hyperbolic metric of the unit ball and the chordal metric are established.