On one extremal problem for complex polynomials with constraints on critical values

Abstract

For all fixed complex numbers a and b and a natural n ≥ 2, we study the problem of finding the supremum of the product |P′(0)P′(1)| over the set of all polynomials P of degree n satisfying the following conditions: P(0) = a and P(1) = b, while |P(z)| ≤ 1 for all z for which P′(z) = 0. As an application of the main result of the article, we give a number of exact estimates for polynomials with account taken of their critical values. We in particular establish a new version of a Markov-type inequality for an arbitrary compact set.