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 Mat. Zametki, 2008, Volume 84, Issue 1, Pages 3–22 (Mi mz5194)

The Sharp Markov–Nikolskii Inequality for Algebraic Polynomials in the Spaces $L_q$ and $L_0$ on a Closed Interval

P. Yu. Glazyrina

Ural State University

Abstract: In this paper, an inequality between the $L_q$-mean of the $k$th derivative of an algebraic polynomial of degree $n\ge 1$ and the $L_0$-mean of the polynomial on a closed interval is obtained. Earlier, the author obtained the best constant in this inequality for $k=0$, $q\in[0,\infty]$ and $1\le k\le n$, $q\in\{0\}\cup[1,\infty]$. Here a new method for finding the best constant for all $0\le k\le n$, $q\in[0,\infty]$, and, in particular, for the case $1\le k\le n$, $q\in(0,1)$, which has not been studied before is proposed. We find the order of growth of the best constant with respect to $n$ as $n\to \infty$ for fixed $k$ and $q$.

Keywords: algebraic polynomial, Markov–Nikolskii inequality, the spaces $L_q$ and $L_0$, geometric mean of a polynomial, $L_q$-mean, extremal polynomial, majorization principle

DOI: https://doi.org/10.4213/mzm5194

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English version:
Mathematical Notes, 2008, 84:1, 3–21

Bibliographic databases:

UDC: 517.518.862

Citation: P. Yu. Glazyrina, “The Sharp Markov–Nikolskii Inequality for Algebraic Polynomials in the Spaces $L_q$ and $L_0$ on a Closed Interval”, Mat. Zametki, 84:1 (2008), 3–22; Math. Notes, 84:1 (2008), 3–21

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz5194
• https://doi.org/10.4213/mzm5194
• http://mi.mathnet.ru/eng/mz/v84/i1/p3

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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. I. E. Simonov, “Tochnoe neravenstvo tipa bratev Markovykh v prostranstvakh $L_p$, $L_1$ na otrezke”, Tr. IMM UrO RAN, 17, no. 3, 2011, 282–290
2. M. R. Gabdullin, “Otsenka srednego geometricheskogo proizvodnoi mnogochlena cherez ego ravnomernuyu normu na otrezke”, Tr. IMM UrO RAN, 18, no. 4, 2012, 153–161
3. V. V. Arestov, M. V. Deikalova, “Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere”, Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 9–23
4. Klurman O., “On Constrained Markov-Nikolskii Type Inequalities For K-Absolutely Monotone Polynomials”, Acta Math. Hung., 143:1 (2014), 13–22
5. Arestov V. Deikalova M., “Nikol'Skii Inequality Between the Uniform Norm and l-Q-Norm With Ultraspherical Weight of Algebraic Polynomials on An Interval”, Comput. Methods Funct. Theory, 15:4, SI (2015), 689–708
6. Sroka G., “Constants in Va Markov'S Inequality in l-P Norms”, J. Approx. Theory, 194 (2015), 27–34
7. Arestov V. Deikalova M., “Nikol'skii inequality between the uniform norm and L q -norm with Jacobi weight of algebraic polynomials on an interval”, Anal. Math., 42:2 (2016), 91–120
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