A Markov-Nikolskii type inequality for absolutely monotone polynomials of order K
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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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