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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 theboundary of $D$. For every $c\geq 2$, $h_{D,c}$ is a metric. In this paper, westudy the sharp Lipschitz constants for the metric $h_{D,c}$ under M\"obiustransformations of the unit ball, the upper half space, and the punctured unitball.

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