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 ( 1 + c d ( x , y ) 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.