Geodesic
Inequalities for harmonic measures with respect to non-overlapping domains
Izvestiya. Mathematics , Tome 79 (2015) no. 5, pp. 902-918.

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Let $B_{1}$ and $B_{2}$ be the components of the complement of a closed Jordan curve $\Gamma\subset\overline{\mathbb{C}}$, and let $E(r)=\{z\colon |z-z_{0}|\leqslant r\}$, where $z_{0}\in\Gamma$. We extend the known inequality for the harmonic measures of $\Gamma\cap E(r)$ with respect to $B_{1}$ and $B_{2}$ to the case of an arbitrary number of pairwise non-overlapping domains $B_{k}$, $k=1,\dots,n$, and prove analogous inequalities for the harmonic measures of sets concentrated in several discs or continua $E_{l}(r)$, $l=1,\dots,m$, of a given logarithmic capacity. We also establish bounds for these measures in terms of the Schwarzian derivatives of functions that conformally map the domains $B_{k}$ onto the unit disc.
Keywords: harmonic measure, condenser capacity, logarithmic capacity, Schwarzian derivative. 
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V. N. Dubinin. Inequalities for harmonic measures with respect to non-overlapping domains. Izvestiya. Mathematics , Tome 79 (2015) no. 5, pp. 902-918. http://geodesic.mathdoc.fr/item/IM2_2015_79_5_a2/

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