Let $E$ be a continuum in the closed unit disk $|z|le 1$ of the complex $z$-plane which divides the open disk $|z| < 1$ into $nge 2$ pairwise non-intersecting simply connected domains $D_k,$ such that each of the domains $D_k$ contains some point $a_k$ on a prescribed circle $|z| = rho, 0 <rho <1, k=1,...,n,. $ It is shown that for some increasing function $Psi,$ independent of $E$ and the choice of the points $a_k,$ the mean value of the harmonic measures $$ Psi^{-1}[ frac{1}{n} sum_{k=1}^{k} Psi(omega(a_k,E, D_k))] $$ is greater than or equal to the harmonic measure $omega(rho, E^*, D^*),,$ where $E^* = {z: z^n in [-1,0] }$ and $D^* ={z: |z|<1, |{rm arg} z| < pi/n} ,.$ This implies, for instance, a solution to a problem of R.W. Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity $inf_{E} max_{k=1,...,n} omega(a_k,E, D_k),$ for arbitrary points of the circle $|z| = rho ,.$ These authors stated this hypothesis in the particular case when the points are equally distributed on the circle $|z| = rho ,.$
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