On Zero Free Regions for Derivatives of a Polynomial

Abstract

. Let P n denote the set of polynomials of the form p ( z ) = ( z − a ) m n − m Y k =1 ( z − z k ) , with | a | ≤ 1 and | z k | ≥ 1 for 1 ≤ k ≤ n − m. For the polynomials of the form p ( z ) = z Q n − 1 k =1 ( z − z k ) , with | z k | ≥ 1, where 1 ≤ k ≤ n − 1, Brown [2] stated the problem “Find the best constant C n such that p 0 ( z ) does not vanish in | z | < C n ”. He also conjectured in the same paper that C n = 1 n . This problem was solved by Aziz and Zarger [1]. In this paper, we obtain the results which generalizes the results of Aziz and Zarger.