Inequalities for critical values of polynomials

Inequalities for the values of an algebraic polynomial P of degree n≥2 at zeros of its derivative P' are obtained. In particular, a problem of Smale is solved: for polynomials of the form P(z)=zn+...+c1z the maximum value of the quantity min{|P(ζ)|:p'(ζ)=0} is found in its dependence on the absolute value of c1. The corresponding proof is based on the dissymmetrization of a certain real function defined on the Riemann surface of the inverse analytic function of the extremal polynomial P*(z)= zn- z.