Roots of a polynomial and its derivatives

Suppose F(z) is a complex polynomial of degree n in one variable. It is known that if two roots of F lie in a disk of radius ρ, then a root of the first derivative F′ lies in a concentric disk of radius O(nρ). We give a generalization: if k+1 roots of F lie in a disk of radius ρ, and l satisfies 1≤l≤k, then at least k+1-l roots of the lth derivative F(l) lie in a disk of radius O((n-k)(k-l)ρ/√k+1), centered at the average of the k+1 roots. We further improve the bound when l = k. We give a lower bound of R = Ω(ρ√n(n-k)/(k+1)) for the case l = k. We give some applications to parallel root finding.