A reflection principle for the hyperbolic metric and applications to geometric function theory

Abstract

We establish a reflection principle for the hyperbolic metric which has applications to geometric function theory. For instance, the reflection principle yields a number of monotonicity properties of the hyperbolic metric. The sharp form of Landau's Theorem is an immediate consequence of one of these monotonicity properties. The second main application is an interpretation of the reflection principle in terms of convexity relative to hyperbolic geometry.