Inequalities for the generalized point pair function

TL;DR

This paper introduces a generalized point pair function, proves it is a quasi-metric for all positive alpha, compares it with existing hyperbolic metrics, and studies its distortion under conformal and quasiregular mappings.

Contribution

It defines a new generalized point pair function, establishes its properties as a quasi-metric, and analyzes its relationships and distortions compared to known metrics.

Findings

The generalized point pair function is a quasi-metric for all alpha > 0.

Derived sharp inequalities relating it to hyperbolic-type metrics.

Analyzed the distortion behavior under conformal and quasiregular mappings.

Abstract

We study a new generalized version of the point pair function defined with a constant $\alpha>0$. We prove that this function is a quasi-metric for all values of $\alpha>0$, and compare it to several hyperbolic-type metrics, such as the $j^*$-metric, the triangular ratio metric, and the hyperbolic metric. Most of the inequalities presented here have the best possible constants in terms of $\alpha$. Furthermore, we research the distortion of the generalized point pair function under conformal and quasiregular mappings.