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Zap. Nauchn. Sem. POMI, 2010, Volume 383, Pages 77–85 (Mi znsl3873)  

This article is cited in 2 scientific papers (total in 2 papers)

On the components of the lemniscate containing no critical points of a polynomial other than its zeros

V. N. Dubinin

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

Abstract: Let $P$ be a complex polynomial of degree $n$ and let $E$ be a connected component of the set $ż\colon|P(z)|\leq1\}$ containing no critical points of $P$ other than its zeros. We prove the inequality $|(z-a)P'(z)/P(z)|\leq n$ for all $z\in E\setminus\{a\}$, where $a$ is the zero of the polynomial $P$ lying in $E$. Equality is attained for $P(z)=cz^n$ and any $z$, $c\neq0$. Bibl. 4 titles.

Key words and phrases: polynomial, lemniscate, Steiner symmetrization.

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English version:
Journal of Mathematical Sciences (New York), 2011, 178:2, 158–162

UDC: 517.54
Received: 17.05.2010

Citation: V. N. Dubinin, “On the components of the lemniscate containing no critical points of a polynomial other than its zeros”, Analytical theory of numbers and theory of functions. Part 25, Zap. Nauchn. Sem. POMI, 383, POMI, St. Petersburg, 2010, 77–85; J. Math. Sci. (N. Y.), 178:2 (2011), 158–162

Citation in format AMSBIB
\Bibitem{Dub10}
\by V.~N.~Dubinin
\paper On the components of the lemniscate containing no critical points of a~polynomial other than its zeros
\inbook Analytical theory of numbers and theory of functions. Part~25
\serial Zap. Nauchn. Sem. POMI
\yr 2010
\vol 383
\pages 77--85
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl3873}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2011
\vol 178
\issue 2
\pages 158--162
\crossref{https://doi.org/10.1007/s10958-011-0534-0}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80053521037}


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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. 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
    2. V. N. Dubinin, A. S. Afanaseva-Grigoreva, “O lemniskatakh ratsionalnykh funktsii”, Dalnevost. matem. zhurn., 17:2 (2017), 201–209  mathnet  elib
  • Записки научных семинаров ПОМИ
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