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Let be a continuum in the closed unit disk of the complex -plane which divides the open disk into pairwise nonintersecting simply connected domains such that each of the domains contains some point on a prescribed circle , It is shown that for some increasing function independent of and the choice of the points the mean value of the harmonic measures is greater than or equal to the harmonic measure where and This implies, for instance, a solution to a problem of RW Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity for arbitrary points of the circle These authors stated this hypothesis in the particular case when the points are equally distributed on the circle References