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Dal'nevostochnyi Matematicheskii Zhurnal, 2004, Volume 5, Number 2, Pages 169–177 (Mi dvmg184)  

This article is cited in 3 scientific papers (total in 4 papers)

Extremal properties of Chebyshev polynomials

V. N. Dubinin, S. I. Kalmykov

Institute of Applied Mathematics, Far-Eastern Branch of the Russian Academy of Sciences
Full-text PDF (201 kB) Citations (4)
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Abstract: Using methods of geometric function theory, we get new extremal properties of Chebyshev polynomials. The exact estimates of coefficients, covering and distortion theorems for polynomials with real coefficients and curved majorants on the interval are obtained. In each case, the extremal is Chebyshov polynomial of second, third or fourth kind. These theorems refine some classical results for algebraic polynomials with constraints on the the interval. As a corollary, we get the following analog of Schur's inequality
$$ \max\{|P(x)|:x\in[-1,1]\}\le(2n+1)\max\{|P(x)\sqrt{(1+x)/2}|:x\in [-1,1]\} $$
where $P(x)$ is the polynomial of degree $n$ with real coefficients. The equality holds for Chebyshev polynomial of the third kind.
Key words: Chebyshev polynomial, polynomial nequality, Bernstein's inequality, Schur's inequality.
Received: 03.07.2004
Document Type: Article
UDC: 512.62, 517.54
MSC: Primary 30C10; Secondary 30C75
Language: Russian
Citation: V. N. Dubinin, S. I. Kalmykov, “Extremal properties of Chebyshev polynomials”, Dal'nevost. Mat. Zh., 5:2 (2004), 169–177
Citation in format AMSBIB
\Bibitem{DubKal04}
\by V.~N.~Dubinin, S.~I.~Kalmykov
\paper Extremal properties of Chebyshev polynomials
\jour Dal'nevost. Mat. Zh.
\yr 2004
\vol 5
\issue 2
\pages 169--177
\mathnet{http://mi.mathnet.ru/dvmg184}
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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