Boundary behaviour of conformal maps. (English) Zbl 0762.30001

Grundlehren der Mathematischen Wissenschaften. 299. Berlin: Springer- Verlag. ix, 300 p. (1992).
The theory of the boundary behavior of conformal maps investigates the connections between the analytic properties of mapping functions and the geometric properties of the image domains. This well-written book gives us a modern presentation of this theory. The book begins with the classical theory of the boundary behavior of conformal maps. Nine of the 11 chapters begin with an overview, where several key results are stated without technical complications. This should make it easy for a non- expert in this area to get a feel for some aspects of this theory. There is a limited number of exercises at the end of most sections, which should make the book useful as a graduate textbook. The 76 figures also contribute greatly to the book. Some chapters (Quasidisks, Smirnov and Lavrentiev domains, Curve families and Capacity) present material in a modern way not yet found in other books. Other chapters (Integral means, Hausdorff measure, Local boundary behavior) present material that can only be found in research papers at this time. The book is not an exhaustive survey (for example, numerical methods and multiply connected domains are not included). The author states in the preface that this personal bias, among other considerations, determined what is included. (The word “fractal” is briefly mentioned as a “set of noninteger dimension”; B. Mandelbrot is neither mentioned nor included in the references).


30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30C85 Capacity and harmonic measure in the complex plane
30C20 Conformal mappings of special domains