A note on a harmonic measure estimate and a conjecture of J. Velling

Suppose that finitely many disjoint open arcs have been selected on the unit circle, each of length less than $\pi$. Let $L_0$ be a longest among them. One can treat the unit disk as a hyperbolic plane in the Poincare disk model. From this perspective each arc $L$ of the selected set determines a hyperbolic half-plane bounded by the geodesic curve $I$ joining endpoints of the arc $L$. Remove from the unit disk all these hyperbolic half-planes. The remaining domain is simply connected and contains the origin. Now map this domain conformally onto the unit disk so that the origin stays fixed. After this map, the boundaries of the hyperbolic half-planes appear as disjoint arcs on the unit circle. Let $I'_0$ be the conformal image of the boundary of the hyperbolic half plane determined by the arc $L_0$. Velling conjectured that $|I'_0| \ge |L_0|,$ where $|\cdot|$ stands for the Euclidean length. He proved some conditional theorems based on validity of this conjecture. In this note we prove a theorem which implies the Velling conjecture and thus converts the conditional theorems of Velling into true ones. 2010 MSC: 30C80, 31A15 Keywords: harmonic measure, extremal problems.