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Keywords:
Lipschitz constant; hyperbolic type metric; Möbius transformation
Lipschitz constant; hyperbolic type metric; Möbius transformation
Summary:
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.
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