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

For all fixed complex numbers $a$ and $b$ and a natural $n\ge2$, 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 $|P(z)|\le1$ 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.