Inequalities for harmonic measures with respect to the nonoverlapping domains

A known inequality for harmonic measures of a part of a closed Jordan curve lying in a disk with respect to the complementary domains is refined. This inequality is extended to an arbitrary number of nonoverlapping domains. We also prove similar inequalities for harmonic measures of sets lying in several disks or continua with a prescribed change of the logarithmic capacities. Moreover, inequalities for such harmonic measures involving the Schwarzian derivatives of the Riemann maps of the nonoverlapping domains are established.