Geodesic
Analogues of the Markov and Bernstein inequalities for polynomials in Banach spaces
Izvestiya. Mathematics , Tome 61 (1997) no. 1, pp. 143-159.

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We propose analogues of the inequalities of A. A. and V. A. Markov for algebraic polynomials on the cube in $\mathbb R^m$ in the $l_2^{(m)}$ and $l_1^{(m)}$ metrics, as well as for polynomials on closed bounded convex bodies in a Banach space and on centrally symmetric bodies of the same type. For the last two cases analogues of Bernstein's inequality are obtained. 
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V. I. Skalyga. Analogues of the Markov and Bernstein inequalities for polynomials in Banach spaces. Izvestiya. Mathematics , Tome 61 (1997) no. 1, pp. 143-159. http://geodesic.mathdoc.fr/item/IM2_1997_61_1_a5/

[1] Markov A. A., “Ob odnom voprose D. I. Mendeleeva”, Izbrannye trudy, GITTL, M.–L., 1948

[2] Markov V. A., O funktsiyakh, naimenee uklonyayuschikhsya ot nulya v dannom promezhutke, SPb., 1892 | Zbl

[3] Bernshtein S. N., Sobranie sochinenii, T. 1, Izd. AN SSSR, M., 1952

[4] Bernshtein S. N., Sobranie sochinenii, T. 2, Izd. AN SSSR, M., 1954

[5] Kellogg O. D., “On bounded polynomials on several variables”, Math. Zeit., 27:1 (1927), 55–66 | DOI | MR

[6] Don R., “Wilhelmsen A. Markov Inequality in Several Dimensions”, J. of Approx. Theory, 11:3 (1974) | MR

[7] Nadzhmiddinov D., Subbotin Yu. N., “Neravenstvo Markova na treugolnikakh”, Matem. zametki, 46:2 (1989), 76–82 | MR | Zbl

[8] Andrianov A. V., “Analogi neravenstv A. Markova i S. Bernshteina dlya mnogochlenov v banakhovykh prostranstvakh”, Matem. zametki, 52:5 (1992), 13–21 | MR | Zbl

[9] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988 | MR | Zbl

[10] Krasnoselskii M. A., Operator sdviga po traektoriyam differentsialnykh uravnenii, Nauka, M., 1966 | MR

[11] Ganzburg M. I., “Teoremy Dzheksona i Bernshteina v ${\mathbb R}^m$”, UMN, 34:1 (1979), 225–226 | MR | Zbl

[12] Schaffer A. S., Duffin R. J., “On some inequalities of S. Bernstein and W. Markoff for derivatives of polynomials”, Bull. Am. Math. Soc., 44 (1938), 289–297 | DOI

[13] Akhiezer N. I., Glazman I. M., Teoriya lineinykh operatorov v gilbertovom prostranstve, Nauka, M., 1966 | MR | Zbl