Inequalities for a new hyperbolic type metric

TL;DR

This paper investigates a new hyperbolic type metric, deriving formulas, inequalities, and bounds, and analyzing its behavior under conformal and quasiregular mappings, enhancing understanding of its geometric properties.

Contribution

It introduces formulas and sharp inequalities for the new metric, improving existing results and analyzing its distortion under various mappings.

Findings

Derived explicit formulas in specific domains

Established sharp inequalities with known metrics

Provided bounds for distortion under conformal mappings

Abstract

We study a new hyperbolic type metric recently introduced by Song and Wang. We present formulas for it in the upper half-space and the unit ball domains and find its sharp inequalities with the hyperbolic metric and the triangular ratio metric. We also improve existing ball inclusion results and give bounds for the distortion of this new metric under conformal and quasiregular mappings.