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 p, then a root of the first derivative F’ lies in a concentric disk of radius O(np). We give a generalization: if k + 1 roots of F lie in a disk of radius p, and C satisfies 1 I t I k, then at least k + 1 C roots of the 4th derivative F(O he in a disk of radius O((n k)(k 4d&-)s centered at the average of the k + 1 roots. We further improve the bound when some abplications to parallel root finding.