Estimates of the harmonic measure of a continuum in the unit disk

FitzGerald, Carl H.; Rodin, Burton; Warschawski, Stefan E.

doi:10.1090/S0002-9947-1985-0768733-1

Estimates of the harmonic measure of a continuum in the unit disk
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by Carl H. FitzGerald, Burton Rodin and Stefan E. Warschawski PDF

Trans. Amer. Math. Soc. 287 (1985), 681-685 Request permission

Abstract:

The harmonic measure of a continuum in the unit disk is estimated from below in two ways. The first estimate is in terms of the angle subtended by the continuum as viewed from the origin. This result is a dual to the Milloux problem. The second estimate is in terms of the diameter of the continuum. This estimate was conjectured earlier as a strengthening of a theorem of D. Gaier. In preparation for the proofs several lemmas are developed. These lemmas describe some properties of the Riemann mapping function of a disk with radial incision onto a disk.

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