The sharp Markov-Nikol'skii inequality for algebraic polynomials in the spaces Lq and L0 on a closed interval

In this paper, an inequality between the L-q-mean of the kth derivative of an algebraic polynomial of degree n >= 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 epsilon [0,infinity] and 1 <= k <= n, q epsilon {0} boolean OR [1,infinity]. Here a newmethod for finding the best constant for all 0 <= k <= n, q epsilon [0,infinity], and, in particular, for the case 1 <= k <= n, q epsilon (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 -> infinity for fixed k and q.