Schwarz pick Type Inequalities

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. The aim of the present book is a uni?ed representation of some recent results in geometric function theory together with a consideration of their historical sources. These results are concerned with functions f, holomorphic or meromorphic in a domain ? in the extended complex planeC. The only additional condition we impose on these functions is the condition that the range f(?) is contained in a given domain ??C.Thisfactwillbedenotedby f? A(?,?). We shall describe (n) how one may get estimates for the derivatives f (z ) ,n?N,f ? A(?,?), 0 dependent on the position of z in ? and f(z)in?. 0 0 1.1 Historical remarks The beginning of this program may be found in the famous article [125] of G. Pick. There, he discusses estimates for the MacLaurin coe?cients of functions with positive real part in the unit disc found by C. Carath' eodory in [52]. Pick tells his readers that he wants to generalize Carath' eodory's estimates such that the special role of the expansion point at the origin is no longer important. For the convenience of our readers we quote this sentence in the original language: Durch lineare Transformation von z oder, wie man sagen darf, durch kreis- ometrische Verallgemeinerung, kann man die Sonderstellung des Wertes z=0 wegscha?en, so dass sich Relationen fur .. die Di?erentialquotienten von w an - liebiger Stelle ergeben. The ?rst great success of this program was G. Pick's theorem, as it is called by Carath' eodory himself, compare [54], vol II, 286-289.