A Markov-type inequality and a lower bound for the moduli of critical values of polynomials

A new version of the Markov-type inequality for the arbitrary plane compact sets is suggested. This inequality arises from a new distortion theorem for the complex polynomials with the constraints on their critical values. Another corollary of this theorem is an exact lower bound for maximal moduli of critical values of polynomials $P$ of degree $n$ normalized by $P(0)=0$.