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

V. N. Dubinin

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 465-482.

Résumé

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.

Détail

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

Bibliographie

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