Abstract:
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.