M\"obius metric in sector domains

TL;DR

This paper investigates the properties of the Möbius metric within sector domains, establishing bounds related to the hyperbolic metric and analyzing its distortion under quasiregular mappings, with numerical insights into polygon domains.

Contribution

It introduces bounds for the Möbius metric in sector domains and explores its distortion under quasiregular mappings, also providing numerical analysis in polygon domains.

Findings

Bounds for the Möbius metric in terms of hyperbolic metric and sector angle

Quantitative estimates of metric distortion under quasiregular mappings

Numerical relationships between Möbius and hyperbolic metrics in polygons

Abstract

The M\"obius metric $\delta_G$ is studied in the cases where its domain $G$ is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the M\"obius metric under quasiregular mappings defined in sector domains. Furthermore, we numerically study the M\"obius metric and its connection to the hyperbolic metric in polygon domains.