The generalized reduced modulus

V. N. Dubinin ; N. V. Eyrikh

Dalʹnevostočnyj matematičeskij žurnal, Tome 3 (2002) no. 2, pp. 150-164.

Résumé

The boundary reduced moduli of the digons and triangles are the essential part of the extremal metric method. They have many applications in the geometric theory of functions of a complex variable. In the present paper, we use the capacity approach to extend these concepts to the concepts of the boundary reduced modulus of the polygons, with any number of vertexs. Moreover, we connect the concept of the boundary reduced modulus with the inner reduced modulus. The correctness of the definition of the generalized reduced modulus is proved. We consider the special cases of reduced modulus, the behavior of the reduced modulus under the extension of the sets and under the conformal mappings of the sets. The principle of the symmetry and the formulae for some reduced moduli are obtained. We prove the new composition principles for the generalized reduced moduli. These principles generalize some theorems about the separating transformation and about the extremal partitioning of the domains.

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