Geodesic
Lipschitz constants for a hyperbolic type metric under Möbius transformations
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 445-460 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $D$ be a nonempty open set in a metric space $(X,d)$ with $\partial D\neq \emptyset $. Define $$ h_{D,c}(x,y)=\log \bigg (1+c\frac {d(x,y)}{\sqrt {d_D(x)d_D(y)}}\bigg ), $$ 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. 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.
Let $D$ be a nonempty open set in a metric space $(X,d)$ with $\partial D\neq \emptyset $. Define $$ h_{D,c}(x,y)=\log \bigg (1+c\frac {d(x,y)}{\sqrt {d_D(x)d_D(y)}}\bigg ), $$ 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. 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.
DOI : 10.21136/CMJ.2024.0366-23
Classification : 30C65, 51M10
Keywords: Lipschitz constant; hyperbolic type metric; Möbius transformation 
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     title = {Lipschitz constants for a hyperbolic type metric under {M\"obius} transformations},
     journal = {Czechoslovak Mathematical Journal},
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Wu, Yinping; Wang, Gendi; Jia, Gaili; Zhang, Xiaohui. Lipschitz constants for a hyperbolic type metric under Möbius transformations. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 445-460. doi: 10.21136/CMJ.2024.0366-23

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