zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $\bbfR^n$, $n\ge 2$. 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.

30C62Quasiconformal mappings in the plane
30C65Quasiconformal mappings in ${\Bbb R}^n$ and other generalizations
33-02Research monographs (special functions)
33-04Machine computation, programs (special functions)
31A15Potentials and capacity, harmonic measure, extremal length (two-dimensional)