Intrinsic metrics defined with arithmetic and logarithmic mean values

TL;DR

This paper introduces new boundary-distance metrics based on arithmetic and logarithmic means, analyzes their properties in various domains, compares them to hyperbolic metrics, and examines their behavior under quasiregular mappings.

Contribution

It defines novel intrinsic metrics using mean values, characterizes when they are true metrics, and explores their relationships and distortions in geometric function theory.

Findings

Identified conditions under which the new functions are metrics.

Established sharp inequalities relating these metrics to hyperbolic metrics.

Analyzed the distortion of these metrics under quasiregular mappings.

Abstract

We introduce several new functions that measure the distance between two points $x$ and $y$ in a domain $G\subsetneq\mathbb{R}^n$ by using the arithmetic or the logarithmic mean of the Euclidean distances from the points $x$ and $y$ to the boundary of $G$. We study in which domains these functions are metrics and find sharp inequalities between them and the hyperbolic metric. We also present one result about their distortion under quasiregular mappings.