Lipschitz constants for a hyperbolic type metric under M\"obius transformations

TL;DR

This paper investigates the precise Lipschitz constants of a hyperbolic-type metric under Möbius transformations across various geometric domains, enhancing understanding of metric distortion in complex analysis.

Contribution

It provides the first sharp bounds for the Lipschitz constants of the metric $h_{D,c}$ under Möbius transformations in key geometric settings.

Findings

Sharp Lipschitz constants are established for the metric under Möbius transformations.

Results apply to the unit ball, upper half space, and punctured unit ball.

The bounds improve understanding of metric distortion in hyperbolic geometry.

Abstract

Let $D$ be a nonempty open set in a metric space $(X,d)$ with $\partial D\neq \emptyset$. Define \begin{equation*} h_{D,c}(x,y)=\log\left(1+c\frac{d(x,y)}{\sqrt{d_D(x)d_D(y)}}\right), \end{equation*} where $d_D(x)=d(x,\partial D)$ is the distance from $x$ to the boundary of $D$. For every $c\geq 2$, $h_{D,c}$ is a metric. In this paper, we study the sharp Lipschitz constants for the metric $h_{D,c}$ under M\"obius transformations of the unit ball, the upper half space, and the punctured unit ball.