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Conformal invariants, inequalities, and quasiconformal maps. (English) Zbl 0885.30012
Canadian Mathematical Society Series of Monographs and Advanced Texts. Chichester: Wiley. xxvii, 505 p. £65.00 (1997).
The principal aim of this book is to give explicit bounds for certain entities in terms of specific functions. The first section, occupying more than one third of the core text, is devoted to assembling known results for these specific functions: hypergeometric, gamma and beta functions, complete elliptic integrals, arithmetic and geometric means, including some results on conformal mapping by elliptic functions. This is followed by a brief chapter on Möbius transformations in ${ℝ}^{n}$, $n\ge 2$. Next comes consideration of conformal invariants starting with the general definition in the context of extremal metrics, then the very special ones on which this book concentrates, primarily the “Grätzsch ring” and Teichmüller’s extremal domain. This is followed by a routine introduction to quasiconformal mappings focussing first on the plane and then on extensions and attempts at extensions to higher dimensions. The text culminates in presenting inequalities for conformal invariants and quasiconformal mappings many of the nature of multiple point distortion theorems. Finally there are numerous appendices, a bibliography and an index, occupying more than one third of the whole book. It is difficult to discern the motivation for publishing a book of this sort, apparantly it is intended to be more than a survey as evinced by the presence of many exercises. However it does not provide an integrated exposition of the central themes. It is unlikely that anyone (except possibly the authors) would use this material as the basis for a lecture course. It is equally unlikely that anyone would read the book through from cover to cover in an organized fashion. Therefore probably it would be useful chiefly in providing a reference to specific results which could be used in a technical manner in research on conformal and quasiconformal mappings.

##### MSC:
 30C62 Quasiconformal mappings in the plane 30C65 Quasiconformal mappings in ${ℝ}^{n}$ and other generalizations 33-02 Research monographs (special functions) 33-04 Machine computation, programs (special functions) 31A15 Potentials and capacity, harmonic measure, extremal length (two-dimensional)