On an extremal problem connected with the harmonic measure (Q1066302)

Let L and q be some positive numbers and let \(E\subset R\) be a measurable set such that for all \(x\in R\) we have \[ mes(E\cap (x,x+L))\leq q. \] Denote by \(\omega\) (z,E) the harmonic measure at the point z, Im z\(>0\), of the set E with respect to the upper half-plane. The authors prove the following estimation \[ (*)\quad \omega (z,E)\leq (2/\pi) \arctan \{cth(\pi Imz/L) \tan (\pi q/2L)\}. \] This estimation is best possible as for \(E=\cup^{\infty}_{k=-\infty}(kL,kL+q)\) we have equality in (*).