Harmonic measure on simply connected domains of fixed inradius

Abstract

LetD⊂C be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. For R>0, let ω_D(R) denote the harmonic measure at 0 of the set {z:|z|≽R}⋔∂D. Then it is shown that there exist β>0and C>0such that for each such D, ω_D(R)≤Ce^−βR, for every R>0. Thus a natural question is: What is the supremum of all β′s , call it β₀, for which the above inequality holds for every such D? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β₀ is found. Upper bounds for β₀ can be obtained by constructing examples of domains D. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.