The problem of extreme decomposition of a complex plane with fixed poles on a circle

Abstract

The problem of extreme decomposition of a complex plane with fixed poles on a circle. Investigation on geometric function theory has been conducted by several researchers, however, few studies have reported on the problem considering extremal configurations the product of inner radii of non-overlapping domains with respect to fixed poles. The paper describes the problem of finding the maximum of the product of inner radii of mutually non-overlapping symmetric domains with respect to points on a unit circle multiply by a certain positive degree \(\gamma\) of the inner radius of the domain with respect to the zero. The problem was studied using the method of separating transformation. Proving the theorem shows that the maximum is obtained if \(\gamma\in(1,n^2]\) and for all \(n\geqslant 2\). Its results and the method for the obtaining of these results can be used in the theory of potential, approximations, holomorphic dynamics, estimation of the distortion problems in conformal mapping, and complex analysis.