Conformal geometry and quasiregular mappings / Matti Vuorinen

This book can perhaps be characterized as a collection of numerous estimates for moduli of path families and capacities as well as of distortion theorems for quasiconformal and quasiregular mappings in Rn. As such, it will serve as a valuable reference.
The book consists of two roughly equally long parts. In the first part (Chapters I and II), the author goes through the basic definitions and a wealth of properties of hyperbolic and quasihyperbolic distances in various domains in Rn, the moduli of path families, capacities of condensers, and some conformal invariants defined in terms of suitable path families. These are to be the tools used in the study of quasiregular maps. The presentation is given with high precision, and references to the original papers are carefully mentioned. All the inequalities and estimates seem to be taken to the extreme, as if to make sure that every possible aspect of a function to be estimated is considered, and that the estimates given are as refined as our current knowledge permits. The author himself has made many contributions to obtaining such refinements. Proofs are carefully written, and many little details are pointed out in the many remarks and exercises. As a result, we get a long collection of formulas and inequalities that cannot be found in any other book.
The second part of the book (Chapters III and IV) deals with quasiregular mappings. A brief survey of the relevant basic definitions and results is given, with no proofs. After that, the tools developed in the first part are used to obtain a great number of estimates for the modulus of continuity of quasiregular maps of the unit ball as well as other domains in Rn. As regards the exposition, the general remarks made above still apply. Chapter IV contains some results on the boundary behavior of quasiregular maps. The book concludes with a list of open problems and an extensive bibliography. The reviewer feels that the book is most suitable for readers who already have some knowledge of quasiconformal or quasiregular mappings and need a comprehensive reference on distortion theorems. (MathSciNet)