Asymptotics of the module of a degenerating condenser and some of their applications

Abstract

The well-known asymptotic formula for the module of a condenser with one of the plates degenerating to a point is generalized to the case of a condenser of general type. The condensers under consideration consist of n plates, n > 2, and the potential functions of condensers take values of different signs on the plates. The asymptotics are considered when one of the plates is fixed while the other n - 1 plates are constracted to points. Applications of the formula to geometric function theory are given. Among them are inequalities for complex numbers and Green functions and also theorems on the extremal decomposition and distortion theorems for univalent functions.