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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 5, Pages 47–64 (Mi izv8313)  

This article is cited in 3 scientific papers (total in 3 papers)

Inequalities for harmonic measures with respect to non-overlapping domains

V. N. Dubininab

a Far Eastern Federal University, Vladivostok
b Institute of Applied Mathematics, Far-Eastern Branch of the Russian Academy of Sciences, Vladivostok

Abstract: Let $B_{1}$ and $B_{2}$ be the components of the complement of a closed Jordan curve $\Gamma\subset\overline{\mathbb{C}}$, and let $E(r)=ż\colon |z-z_{0}|\leqslant r\}$, 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 $B_{k}$, $k=1,…,n$, and prove analogous inequalities for the harmonic measures of sets concentrated in several discs or continua $E_{l}(r)$, $l=1,…,m$, 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.

Funding Agency Grant Number
Russian Science Foundation 14-11-00022
This work is supported by the Russian Science Foundation under grant no. 14-11-00022.


DOI: https://doi.org/10.4213/im8313

Full text: PDF file (545 kB)
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English version:
Izvestiya: Mathematics, 2015, 79:5, 902–918

Bibliographic databases:

Document Type: Article
UDC: 517.54
MSC: 30C85, 31A15, 30C75
Received: 06.11.2014
Revised: 23.06.2015

Citation: V. N. Dubinin, “Inequalities for harmonic measures with respect to non-overlapping domains”, Izv. RAN. Ser. Mat., 79:5 (2015), 47–64; Izv. Math., 79:5 (2015), 902–918

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. F. G. Avkhadiev, P. L. Shabalin, “Conformal mappings of circular domains on finitely-connected non-Smirnov type domains”, Ufa Math. J., 9:1 (2017), 3–17  mathnet  crossref  isi  elib
    2. V. N. Dubinin, “Geometric estimates for the Schwarzian derivative”, Russian Math. Surveys, 72:3 (2017), 479–511  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. Dubinin V.N., “Some Unsolved Problems About Condenser Capacities on the Plane”, Complex Analysis and Dynamical Systems: New Trends and Open Problems, Trends in Mathematics, eds. Agranovsky M., Golberg A., Jacobzon F., Shoikhet D., Zalcman L., Birkhauser Verlag Ag, 2018, 81–92  crossref  mathscinet  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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