Schwarz-Pick Type Inequalities: Frontiers in Mathematics

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

From the reviews:
“The aim of this book is to give a unified presentation of some recent results in geometric function theory together with a consideration of their historical sources. The extensive historical references are … interesting, thorough and informative. … this book is filled with many challenging conjectures and suggested problems for exploring new research. In summary this is a delightful book that anyone interested in interrelating geometry and classical geometric function theory should read.”­­­ (Roger W. Barnard, Mathematical Reviews, Issue 2010 j)

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