Abstract
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 ε I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in L p [−1, 1], p > 0, is established. One may guess that the right Markov factor is cn 2/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.
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Dedicated to the memory of Professor George G. Lorentz on the occasion of his 100th birthday
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Erdélyi, T. A Markov-Nikolskii type inequality for absolutely monotone polynomials of order K . JAMA 112, 369–381 (2010). https://doi.org/10.1007/s11854-010-0034-z
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DOI: https://doi.org/10.1007/s11854-010-0034-z