The symmetrization method in problems on nonoverlapping domains

A new approach to the use of symmetrization is considered. Sterner symmetrization is taken as the main tool. An arbitrary symmetrization transformation connected with a given quadratic differential $Q(z)dz^2$ is obtained by successive application of the mappng $\zeta=\int Q^{1/2}(z)\,dz$ and Steiner symmetrization.
As a consequence of the main theorem, the corresponding results of Lavrent'ev, Goluzin, Jenkins, and others are refined and generalized to the case of domains of arbitrary connectivity (not necessarily having a filling).
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