Let $D$ be a nonempty open set in a metric space $(X,d)$ with $\partial D\ne \varnothing$. Define $$h_{D,c}(x,y)=log\left(1+c\frac{d(x,y)}{\sqrt{d_D\left(x\right)d_D\left(y\right)}}\right),$$ where $d_D\left(x\right)=d(x,\partial D)$ is the distance from $x$ to the boundary of $D$. For every $c\ge 2$, $h_{D,c}$ 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.