A 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 this indeed turns out to be the case. Similarly sharp results hold in the case of higher derivatives and Markov-Nikolskii type inequalities. There is also 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 the paper. 2010 Hebrew University Magnes Press.

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