Let and be two simply connected domains in the complex plane which are not equal to . We denote by and the densities of the hyperbolic metrics in and respectively. Let denote the set of functions analytic in . The authors consider the quantities
It follows by the Schwarz-Pick lemma that for any pair of simply connected domains. The quantity has been computed in some special cases by St. Ruscheweyh (for example, when is the unit disk and is a half-plane).
The authors of the article under review show that for any convex domain ,
It follows that for convex domain . Furthermore, they show that holds for arbitrary simply connected domains whereas the inequality is proved only under some technical restriction upon and .