The Schwarz Lemma

The Schwarz lemma is among the simplest results in complex analysis that capture the rigidity of holomorphic functions. This self-contained volume provides a thorough overview of the subject; it assumes no knowledge of intrinsic metrics and aims for the main results, introducing notation, secondary concepts, and techniques as necessary. Suitable for advanced undergraduates and graduate students of mathematics, the two-part treatment covers basic theory and applications.
Starting with an exploration of the subject in terms of holomorphic and subharmonic functions, the treatment proves a Schwarz lemma for plurisubharmonic functions and discusses the basic properties of the Poincaré distance and the Schwarz-Pick systems of pseudodistances. Additional topics include hyperbolic manifolds, special domains, pseudometrics defined using the (complex) Green function, holomorphic curvature, and the algebraic metric of Harris. The second part explores fixed point theorems and the analytic Radon-Nikodym property.

Sean Dineen is Professor Emeritus of Mathematics at the University College Dublin School of Mathematical Sciences. His other books include Complex Analysis of Infinite Dimensional Spaces, Complex Analysis in Locally Convex Spaces, and Probability Theory in Finance: A Mathematical Guide to the Black-Scholes Formula.

PART I1. The classical Schwarz lemma2. A Schwarz lemma for plurisubharmonic functions3. The Poincaré distance on the unit disc4. Schwarz-Pick systems of pseudodistances5. Hyperbolic manifolds6. Special domains7. Pseudometrics defined using the (complex) Green function8. Holomorphic curvature9. The algebraic metric of HarrisPART II10. A holomorphic characterization of Banach spaces containing c011. Fixed point theorems12. The analytic Radon-Nikodym propertyReferencesIndex