On one extremal problem for complex polynomials with constraints on critical values (Q467659)

Let \(P_n\) be the set of all complex polynomials \(P\) of degree \(n\geq 2\) satisfying \(|P(z)|\leq 1\) whenever \(P'(z)=0.\) Using a symmetrisation method, the author proves a result which gives an upper bound for the product \(|P'(0)P'(1)|\), \(P\in P_n\), in terms of certain functions of the values \(|P(0)|\) and \(|P(1)|\). As an application of this result, a number of sharp inequalities for polynomials is presented. In particular, a new version of a Markov-type inequality for an arbitrary compact set is established.