Lipschitz constants for a hyperbolic type metric under Möbius transformations

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