Hyperbolic geometric versions of Schwarz’s lemma

Abstract

Let f f be a holomorphic self-map of the unit disk D \mathbb {D} . We prove monotonicity theorems which involve the hyperbolic area, the hyperbolic capacity, and the hyperbolic diameter of the images under f f of hyperbolic disks in D \mathbb {D} . These theorems lead to distortion and modulus growth theorems that generalize the classical lemma of Schwarz and to geometric estimates for the density of the hyperbolic metric.