Harmonic measure on simply connected domains of fixed inradius

Dimitrios Betsakos Anatolikis Romilias 7, Kozani, Greece

TBD mathscidoc: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.
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  title={Harmonic measure on simply connected domains of fixed inradius},
  author={Dimitrios Betsakos},
  booktitle={Arkiv for Matematik},
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.
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