Triangular ratio metric in the unit disk

TL;DR

This paper investigates the properties of the triangular ratio metric within the unit disk, establishing sharp bounds and exploring its implications for the regularity of quasiconformal mappings.

Contribution

It provides new sharp bounds for the triangular ratio metric in the unit disk and applies these results to analyze the H"older continuity of quasiconformal mappings.

Findings

Sharp bounds for the triangular ratio metric in the unit disk

Application to H"older continuity of quasiconformal mappings

Enhanced understanding of metric behavior in complex domains

Abstract

The triangular ratio metric is studied in a domain $G\subsetneq\mathbb{R}^n$, $n\geq2$. Several sharp bounds are proven for this metric, especially, in the case where the domain is the unit disk of the complex plane. The results are applied to study the H\"older continuity of quasiconformal mappings.