Inequalities for critical values of polynomials

Inequalities for the values of an algebraic polynomial $P$ of degree $n\geqslant2$ at zeros of its derivative $P'$ are obtained. In particular, a problem of Smale is solved: for polynomials of the form $P(z)=z^n+\dots+c_1z$ the maximum value of the quantity $\min\{|P(\zeta)|:P'(\zeta)=0\}$ is found in its dependence on the absolute value of $c_1$. 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 $P^*(z)=z^n-z$.
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