An extremal decomposition problem for harmonic measure

Let $E$ be a continuum in the closed unit disk $\vert z\vert\le 1$ of the complex $z$-plane which divides the open disk $\vert z\vert < 1$ into $n\ge 2$ pairwise nonintersecting simply connected domains $D_k$ such that each of the domains $D_k$ contains some point $a_k$ on a prescribed circle $\vert z\vert = \rho$, $0 <\rho <1\, , \, k=1,\dots ,n\,.$ 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

$$\Psi ^{-1}\left [ \frac {1}{n} \sum _{k=1}^{k} \Psi ( \omega (a_k,E, D_k))\right ]$$

is greater than or equal to the harmonic measure $\omega (\rho , E^* , D^*)\,,$ where $E^* = \{ z: z^n \in [-1,0] \}$ and $D^* =\{ z: \vert z\vert<1, \vert{\rm arg}\, z\vert < \pi /n\} \,.$ 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 $\inf _{E} \max _{k=1,\dots ,n} \omega (a_k,E, D_k)\,$ for arbitrary points of the circle $\vert z\vert = \rho \,.$ These authors stated this hypothesis in the particular case when the points are equally distributed on the circle $\vert z\vert = \rho \,.$