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Two-sided bounds for the logarithmic capacity of multiple intervals

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Abstract

Potential theory on the complement of a subset of the real axis attracts much attention in both function theory and applied sciences. This paper discusses one aspect of the theory — the logarithmic capacity of closed subsets of the real line. We give simple but precise upper and lower bounds for the logarithmic capacity of multiple intervals and a lower bound valid also for closed sets comprising an infinite number of intervals. Using some known methods to compute the exact values of capacity, we demonstrate graphically how our estimates compare with them. The main machinery behind our results are the separating transformation and dissymmetrization developed by V. N. Dubinin and a version of the latter due to K. Haliste, as well as some classical symmetrization and projection results for the logarithmic capacity. The results of the paper improve some previous achievements by A. Yu. Solynin and K. Shiefermayr.

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Authors and Affiliations

  1. Institute of Appliedmathematics, Far Eastern Branch of the Russian Academy of Sciences, 7 Radio Street, Vladivostok, 690041, Russia

    V. N. Dubinin & D. Karp

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  1. V. N. Dubinin
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  2. D. Karp
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Correspondence to V. N. Dubinin.

Additional information

This work is supported by the Far Eastern Branch of the Russian Academy of Sciences (grants 09-III-A-01-008 and 09-II-CO-01-003), the Russian Basic Research Fund (grant 08-01-00028-a) and the Russian Federal Program “Scientific and Pedagogical Personnel of Innovative Russia” (grant 02.740.11.0198).

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Dubinin, V.N., Karp, D. Two-sided bounds for the logarithmic capacity of multiple intervals. JAMA 113, 227–239 (2011). https://doi.org/10.1007/s11854-011-0005-z

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  • DOI: https://doi.org/10.1007/s11854-011-0005-z

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