Czechoslovak Mathematical Journal, first online, pp. 1-16
Lipschitz constants for a hyperbolic type metric under Möbius transformations
Yinping Wu, Gendi Wang, Gaili Jia, Xiaohui Zhang
Received August 9, 2023. Published online February 12, 2024.
Abstract: Let $D$ be a nonempty open set in a metric space $(X,d)$ with $\partial D\neq\emptyset$. Define $h_{D,c} (x,y)=\log (1+c\frac{d(x,y)}{\sqrt{d_D(x)d_D(y)}}),$ where $d_D(x)=d(x,\partial D)$ is the distance from $x$ to the boundary of $D$. For every $c\geq2$, $h_{D,c}$ is a metric. We study the sharp Lipschitz constants for the metric $h_{D,c}$ under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.
Keywords: Lipschitz constant; hyperbolic type metric; Möbius transformation
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