SMALE’S PROBLEM FOR CRITICAL POINTS ON CERTAIN TWO RAYS

Source abstract

Abstract 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 ∣ f ( ζ )/ ζ ∣<2/3 if they lie in the union of the two rays {1+ re ± iθ : r ≥0}, where 0< θ ≤ π /2.