Polynomials with Critical Values on Intervals

For polynomials $P(z)$ with real coefficients having a fixed leading coefficient and satisfying the conditions $P(z)\in[-1,1]$ for $z\in[-1,1]$ and $P(z)\in[-1,1]$ if $P'(z)=0$, we obtain new covering theorems, a Bernshtein-type inequality, and inequalities for the coefficients. The proofs are based on the use of univalent conformal mappings.