Univalent Functions. With a Chapter on Quadratic Differentials by Gerd Jensen. (Studia mathematica Band XXV) - ISBN:3525401337

Univalent Functions. With a Chapter on Quadratic Differentials by Gerd Jensen. (Studia mathematica Band XXV)

Date: 11 June 2011, 02:34

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Every simply connected domain in the extended complex plane can be mapped conformally onto a standard domain such as the unit disk. This book develops the theory of univalent (= injective meromorphic) functions in the unit disk (or in the exterior of the unit circle if this is more convenient).
The first part deals with the various methods developed to study extremal problems such as the Bieberbach conjecture. Our coverage of methods is reasonably complete; we omit the symmetrization method (see Hayman Chapt. 4 or Jenkins Chapt. 8) and a closer study of the coefficient body (see Schaeffer and Spencer). Of course, we can prove only a part of the many results on univalent functions. We do not consider multiply connected domains nor multivalent functions. The chapter on quadratic differentials, written by Gerd Jensen, contains a proof of Jenkins' General Coefficient Theorem.
The second part of the book deals with the boundary behaviour of univalent functions. The author's bias in the choice of material ist quite obvious. The development is based on the theory of normal functions. We hardly mention some important subjects such as harmonic measure and extremal length on which there are excellent books, for instance Nevanlinna, Jenkins and Ahlfors (Conf. Inv.). Quasiconformal mappings are only briefly studied.
The material, in particular in the first part, is arranged according to method and not to results. Thus we sometimes prove the same result by several different methods.

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