MARKOV-NIKOLSKII TYPE INEQUALITY FOR ABSOLUTELY MONOTONE POLYNOMIALS OF ORDER k

A function Q is called absolutely monotone of order k on an interval I if Q(x) ≥ 0, Q(x) ≥ 0, . . . , Q(k)(x) ≥ 0, for all x ∈ I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in Lp[−1, 1], p > 0, is established. One may guess that the right Markov factor is cn2/k and, indeed, this turns out to be the case. Moreover, similarly sharp results hold in the case of higher derivatives and Markov-Nikolskii type inequalities. There is a remarkable connection between the right Markov inequality for absolutely monotone polynomials of order k in the supremum norm, and essentially sharp bounds for the largest and smallest zeros of Jacobi polynomials. This is discussed in the last section of this paper.