An extremal decomposition problem for harmonic measure

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 $n\ge 2$ 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 $|z| = \rho \,.$