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Mat. Zametki, 2006, Volume 80, Issue 1, Pages 33–37 (Mi mz2777)  

This article is cited in 5 scientific papers (total in 5 papers)

Lemniscates and Inequalities for the Logarithmic Capacities of Continua

V. N. Dubinin

Institute of Applied Mathematics, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: It is shown that if $P(z)=z^n+\dotsb$ is a polynomial with connected lemniscate $E(P)=ż\colon |P(z)|\le 1\}$ and $m$ critical points, then, for any $n-m+1$ points on the lemniscate $E(P)$, there exists a continuum $\gamma\subset E(P)$ of logarithmic capacity $\operatorname{cap}\gamma\le 2^{-1/n}$ which contains these points and all zeros and critical points of the polynomial. As corollaries, estimates for continua of minimum capacity containing given points are obtained.

Keywords: lemniscate of a polynomial, logarithmic capacity, continuum

DOI: https://doi.org/10.4213/mzm2777

Full text: PDF file (428 kB)
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English version:
Mathematical Notes, 2006, 80:1, 31–35

Bibliographic databases:

UDC: 512.62+517.54
Received: 01.12.2005

Citation: V. N. Dubinin, “Lemniscates and Inequalities for the Logarithmic Capacities of Continua”, Mat. Zametki, 80:1 (2006), 33–37; Math. Notes, 80:1 (2006), 31–35

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. N. Dubinin, D. B. Karp, V. A. Shlyk, “Izbrannye zadachi geometricheskoi teorii funktsii i teorii potentsiala”, Dalnevost. matem. zhurn., 8:1 (2008), 46–95  mathnet  elib
    2. V. V. Vasin, V. N. Dubinin, V. G. Romanov, “Itogovyi nauchnyi otchet po mezhdistsiplinarnomu integratsionnomu proektu SO RAN: “Razrabotka teorii i vychislitelnoi tekhnologii resheniya obratnykh i ekstremalnykh zadach s prilozheniem v matematicheskoi fizike i gravimagnitorazvedke””, Sib. elektron. matem. izv., 5 (2008), 427–439  mathnet  elib
    3. V. N. Dubinin, “Emkosti kondensatorov i printsipy mazhoratsii v geometricheskoi teorii funktsii kompleksnogo peremennogo [Itogovyi nauchnyi otchet po mezhdistsiplinarnomu integratsionnomu proektu SO RAN: “Razrabotka teorii i vychislitelnoi tekhnologii resheniya obratnykh i ekstremalnykh zadach s prilozheniem v matematicheskoi fizike i gravimagnitorazvedke”]”, Sib. elektron. matem. izv., 5 (2008), 465–482  mathnet  mathscinet
    4. V. N. Dubinin, “Methods of geometric function theory in classical and modern problems for polynomials”, Russian Math. Surveys, 67:4 (2012), 599–684  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. V. N. Dubinin, A. S. Afanaseva-Grigoreva, “O lemniskatakh ratsionalnykh funktsii”, Dalnevost. matem. zhurn., 17:2 (2017), 201–209  mathnet  elib
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