This book discusses in detail the extension of the Schwarz-Pick inequality to higher order derivatives of analytic functions with given images. It is the first systematic account of the main results in this area. RecentВ resultsВ in geometric function theory presented here include the attractive steps on coefficient problems from Bieberbach to de Branges, applications of some hyperbolic characteristics of domains via Beardon-Pommerenke's theorem, a new interpretation of coefficient estimates as certain properties of the PoincarГ© metric, and a successful combination of the classical ideas of Littlewood and TeichmГјller with modern approaches.В The material is complemented with historical remarks on the Schwarz Lemma and a chapterВ introducing some challengingВ open problems.

The book will be of interest for researchers and postgraduate students in function theory and hyperbolic geometry.