Harmonic measure on simply connected domains of fixed inradius

Let D subset of C be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. For R>0, let omega(D)(R) denote the harmonic measure at 0 of the set {z:\z\greater than or equal to R}boolean AND partial derivative D. Then it is shown that there exist beta>0 and C>0 such that for each such D, omega(D)(R)less than or equal to Ce-beta R, for every R>0. Thus a natural question is: What is the supremum of all beta's, call it Pot 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 beta(0) is found. Upper bounds for beta(0) 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.