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arXiv · 2309.03515

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

Yinping Wu·Gendi Wang·Gaili Jia·Xiaohui Zhang
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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.

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Yinping Wu, Gendi Wang, Gaili Jia, Xiaohui Zhang. 2023-09-07. Lipschitz constants for a hyperbolic type metric under M\"obius transformations. https://arxiv.org/abs/2309.03515

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