Generalization of the Smirnov Operator and Differential Inequalities for Polynomials

Abstract

The question raised in this article goes back to the problem posed by the famous chemist D. I. Mendeleev in 1887 (solved by A. A. Markov in 1889). In the next 100 years, the Mendeleev problem was repeatedly modificated and solved. Its essence is in the description of conditions under which the inequality ∣f(z)∣ ≤ ∣F(z)∣ for polynomials f and F and for z from a fixed set implies the inequality ∣L[f](z)∣ ≤ ∣L[F](z)∣ for some differential operator L. In the presented paper, we consider a differential operator of special type and arbitrary order. In particular, we obtain a sharp upper estimate for higher order derivatives of arbitrary polynomial in terms of the polynomial values.