# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.332897

Arkiv for Matematik, 36, (2), 275-306, 1996.5
Let\$D\$⊂\$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>0\$and C\$>0\$such 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 β_{0}, 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 β_{0}is found. Upper bounds for β_{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.
```@inproceedings{dimitrios1996harmonic,
title={Harmonic measure on simply connected domains of fixed inradius},
author={Dimitrios Betsakos},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203553725706706},
booktitle={Arkiv for Matematik},
volume={36},
number={2},
pages={275-306},
year={1996},
}
```
Dimitrios Betsakos. Harmonic measure on simply connected domains of fixed inradius. 1996. Vol. 36. In Arkiv for Matematik. pp.275-306. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203553725706706.