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Let E be a continuum in the closed unit disk| z|≤ 1 of the complex z-plane which divides the open disk| z|< 1 into n≥ 2 pairwise non-intersecting simply connected domains Dk, such that each of the domains Dk contains some point ak on a prescribed circle| z|= ρ, 0< ρ< 1, k= 1,..., n. It is shown that for some increasing function Ψ, independent of E and the choice of the points ak, the mean value of the harmonic measures