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Geometric function theory. Explorations in complex analysis. (English) Zbl 1086.30001

Cornerstones. Boston, MA: Birkhäuser (ISBN 0-8176-4339-7/hbk). xiii, 314 p. (2006).
The geometric point of view is the unifying theme in this fine textbook in complex function theory. But the author also studies byways that come from analysis and algebra. In fact, on one hand he treats invariant geometry, the Bergman metric, the automorphism groups of domains and the boundary regularity of conformal mappings, and on the other hand, he explores the Hilbert transform, the Laplacian, the corona theorem, harmonic measure, the inhomogeneous Cauchy-Riemann equations and sheaf theory. Altogether, the author treats advanced topics that lead the reader to modern areas of research. And what is important, the topics are presented with an explanation of their interaction with other important parts of mathematics. The presentations of the topics are clear and the text makes a very good reading; basic ideas of many concepts and proofs are carefully described, non-formal introductions to each chapter are very helpful, a rich collection of exercises is well composed and helps the student to understand the subject. The book under review leads the student to see what complex function theory has to offer and thereby give him or her a taste of some of the areas of current research. As such it is a welcome addition to the existing literature in complex function theory.
The book is divided into three parts. Each part begins with an overview indicating the contents of the following chapters. Each chapter is accomplished by a set of problems for study and exploration. Part I of the book is entitled Classical Function Theory and consists of six chapters: Invariant Geometry; Variations on the Theme of the Schwarz Lemma; Normal families; The Riemann Mapping Theorem and Its Generalizations; Boundary Regularity of Conformal Maps; The Boundary behaviour of Holomorphic Functions. Part II deals with Real and Harmonic Analysis and has five chapters: The Cauchy-Riemann Equations; The Green’s function and the Poisson Kernel; Harmonic Measure; Conjugate Functions and The Hilbert Transform; The Wolff’s Proof of the Corona Theorem. The last part is called Algebraic Topics and consists of two chapters: Automorphism Groups of Domains in the plane; Cousin Problems, Cohomology, and Sheaves.
In this reviewer’s opinion, the book can warmly be recommended both to experts and to a new generation of mathematicians.

MSC:

30-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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