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 Sib. Èlektron. Mat. Izv., 2008, Volume 5, Pages 465–482 (Mi semr121)

Reviews

Condenser capacities and majorization principles in the geometric function theory of a complex variable

V. N. Dubinin

Institute of Applied Mathematics, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: This survey paper is devoted to applications of potential theory to some extremal problems of the geometric function theory of a complex variable. In particular, we present variational principles of conformal mappings that are derived from the properties of generalized condensers and symmetrization in a unified way. The variations of the Robin functions under deformation of a domain or a portion of its boundary are considered. Applications of condensers and majorization principles include distortion theorems for holomorphic functions, covering theorem for $p$-valent functions in a circular annulus, Bernstein-type inequalities for rational functions with prescribed poles, polynomial inequalities and more.

Keywords: Condenser capacity, hyperbolic capacity, logarithmic capacity, Robin function, symmetrization, dissimmetrization, variational principles, majorization principles, conformal mappings, distortion theorems, covering theorems, $p$-valent functions, rational functions, polynomials.

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Bibliographic databases:
UDC: 512.62, 517.54, 517.956
MSC: 30C10, 30C25, 30C75, 30C85, 31A15
Received September 1, 2008, published November 26, 2008

Citation: V. N. Dubinin, “Condenser capacities and majorization principles in the geometric function theory of a complex variable”, Sib. Èlektron. Mat. Izv., 5 (2008), 465–482

Citation in format AMSBIB
\Bibitem{Dub08} \by V.~N.~Dubinin \paper Condenser capacities and majorization principles in the geometric function theory of a~complex variable \jour Sib. \Elektron. Mat. Izv. \yr 2008 \vol 5 \pages 465--482 \mathnet{http://mi.mathnet.ru/semr121} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2586651} `