Sib. Èlektron. Mat. Izv., 2008, Volume 5, Pages 465–482
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
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.
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.
PDF file (830 kB)
512.62, 517.54, 517.956
MSC: 30C10, 30C25, 30C75, 30C85, 31A15
Received September 1, 2008, published November 26, 2008
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
\paper Condenser capacities and majorization principles in the geometric function theory of a~complex variable
\jour Sib. \`Elektron. Mat. Izv.
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