  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.
