For polynomials P( z) with real coefficients having a fixed leading coefficient and satisfying the conditions P( z) ∈ [−1, 1] for z ∈ [−1, 1] and P( z) ∈ [−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.