An extremal decomposition problem for harmonic measure

Abstract

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 $$ Ψ^{-1}\[ \frac{1}{n} \sum_{k=1}^{k} Ψ(ω(a_k,E, D_k))] $$ is greater than or equal to the harmonic measure $\omega (\rho ,E^{\ast },D^{\ast }),$ where $E^{\ast }=\{z:z^n\in [-1,0]\}$ and $D^{\ast }=\{z:|z|<1,|\arg z|<\pi /n\}.$ 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 $|z|=\rho .$

6 pages, 2 figures