The distance ratio metric on some parts of $\mathbb C$ plane
  We prove a generalization for some $\mathbb C$ domains of Gehring - Palka theorem on Moebius transformations regarding the distance ratio metric. Namely, we show that this theorem is valid for arbitrary holomorphic mappings $f : H \to H$ or $f : D \to D$ with the same Lipschitz constant $2$. 
Slavko Simic
math.CV