Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that {pipe} f(ζ)/ζ{pipe} ≤ 2/3, and that this inequality can be taken ...

Let f be a polynomial of degree n ≥ 2 with f(0)= 0 and f′(0)= 1. We prove that there is a critical point ζ of f with |f(ζ)/ζ| ≤ 1/2 provided that the critical points of f lie in the sector {re iθ:r > 0,|θ| ≤ π/6}, and ...