The sharp Markov-Nikol’skii inequality for algebraic polynomials in the spaces L q and L 0 on a closed interval

Abstract

In this paper, an inequality between the Lq-mean of the kth derivative of an algebraic polynomial of degree n ≥ 1 and the L0-mean of the polynomial on a closed interval is obtained. Earlier, the author obtained the best constant in this inequality for k = 0, q ∈ [0,∞] and 1 ≤ k ≤ n, q ∈ {0} ∪ [1,∞]. Here a newmethod for finding the best constant for all 0 ≤ k ≤ n, q ∈ [0,∞], and, in particular, for the case 1 ≤ k ≤ n, q ∈ (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 → ∞ for fixed k and q.