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

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.