On Constrained Markov–Nikolskii type Inequalities for k-Absolutely Monotone Polynomials

Abstract

We consider a classical problem of estimating norms of higher order derivatives of an algebraic polynomial via the norms of the polynomial itself. The corresponding extremal problem for general polynomials in the uniform norm was solved by V. A. Markov. In 1926, Bernstein found the exact constant in the Markov inequality for monotone polynomials. It was shown in [3] that the order of the constants in constrained Markov–Nikolskii inequality for k-absolutely monotone polynomials is the same as in the classical one in case \({0 < p \leqq q \leqq \infty}\). In this paper, we find the exact order for all values of \({0 < p, q \leqq \infty}\). It turnes out that for the case q < p the constrained Markov–Nikolskii inequality is significantly better than the unconstrained one.