Inequalities for polynomials with real critical values

Under certain geometrical constraints on the polynomials with real critical values, we give lower and upper estimates of the moduli of the polynomial and of its derivative independent of the degree of this polynomial. As a corollary, we refine Hinkkanen and Kayumov's estimate for min\|P(ζ)/ζ| P'(ζ)=0\ for polynomials P, $P(0)=0$, $P'(0)=1$, with real critical points.