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Symmetrization in the geometric theory of functions of a complex variable. (English. Russian original) Zbl 0830.30014
Russ. Math. Surv. 49, No. 1, 1-79 (1994); translation from Usp. Mat. Nauk 49, No. 1(295), 3-76 (1994).
Already in the introduction the author presents general properties inhering in symmetrization. Let $$U$$ denote an abstract set and let $$F(u)$$ be a real function on $$U$$. The following extremal problem is considered: (1) $$F(u) \to \inf$$, $$u \in U$$. At the basis of the symmetrization method lies the idea that any extremal configuration should distinguish itself by some symmetry and, in consequence, the solution of the given problem should be sought for among objects possessing elements of this symmetry. A priori, the symmetry of extremal objects is not known in general. The problem of symmetrization consists in defining a proper subset $$U^* \subset U$$ of elements $$u$$ having some symmetry and in constructing the mapping Sym: $$U \to U^*$$ so that (2) $$F(u) \geq F (\text{Sym} u)$$, $$u \in U$$. If such a mapping exists, then problem (1) reduces to $$F(u) \to \inf$$, $$u \in U^*$$. The mapping Sym is called a symmetrization, and inequality (2) – the symmetrization principle. The symmetrization method consists of the symmetrization principles corresponding to various Sym. The symmetrization method has found and still finds effective applications in the geometrical theory of complex functions. These applications develop in the following two directions: 1) the constructing of new geometrical transformations Sym for which symmetrization principle (2) holds, 2) the defining of new functions $$F(u)$$ satisfying symmetrization principle and the investigating of the properties of these functions, connected with certain classes of conformal mappings. The paper being reviewed is, in substance, devoted to a survey of the results within the scope of 1) and their applications. The paper distinguishes itself by the richness of the results, the clarity of presenting them, as well as a rich list of references. It also contains the formulations of several open problems.

##### MSC:
 30C99 Geometric function theory
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