This book gives a unified representation of generalizations of the Schwarz Lemma. It examines key coefficient theorems of the last century and explains the connection between coefficient estimates and characteristics of the hyperbolic geometry in a domain.

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. Recent results 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 Poincaré metric, and a successful combination of the classical ideas of Littlewood, Löwner and Teichmüller with modern approaches. The material is complemented with historical remarks on the Schwarz Lemma and a chapter introducing some challenging open problems.

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

Contains historical remarks on the Schwarz Lemma as well as new theorems on Schwarz-Pick inequalities from the last 25 years In addition to the several analytic methods, readers will find many interesting applications of geometric properties of domains from very special cases to domains with uniformly perfect boundary A lot of open problems, old and new, in geometric function theory are discussed in detail Includes supplementary material: sn.pub/extras

Inhalt
Basic coefficient inequalities.- The Poincaré metric.- Basic Schwarz-Pick type inequalities.- Punishing factors for special cases.- Multiply connected domains.- Related results.- Some open problems.