Schwarz Lemma

Intrinsic metrics on manifolds provide a means to present a unified account of analysis in one, several, and infinitely many complex variables. In this book, Professor Dineen develops this theme and demonstrates the utility of this approach to central problems in complex analysis. Prerequisites are little more than a knowledge of basic one variable complex analysis, together with some functional analysis. Consequently the book will be suitable for both graduate students and research workers coming to these topics for the first time. The first part of the book is devoted to generalizations of Poincare, Harris, Kobayashi metrics, and pseudometrics on domains. The latter half of the book develops more advanced topics such as k-hyperbolic domains, H P -spaces, and applications to Banach algebra theory and fixed point theorems applied to bounded subsets of Banach spaces and holomorphic mappings.Table of ContentPART I: The classical Schwarz lemma; A Schwarz lemma for plurisubharmonic functions; The Poincare distance on the unit disc; Schwarz-Pick systems of pseudodistances; Hyperbolic manifolds; Special domains; Pseudometrics defined using the (complex) green function; Holomorphic curvature; The algebraic metric of Harris; PART II: A holomorphic characterization of Banch spaces containing CO]; Fixed point theorems; The analytic Radon-Nikodym property; References; Index