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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     title = {Analogues of the {Markov} and {Bernstein} inequalities for polynomials in {Banach} spaces},
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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/