On zero free regions for derivatives of a polynomial

Let Pn denote the set of polynomials of the form p(z) = (z - a)m n- m ri k=1 (z - zk), with |a| <= 1 and |zk| >= 1 for 1 <= k <= n - m. For the polynomials of the form p(z) = z pi n-1k=1 (z - zk), with |zk| >= 1, where 1 <= k <= n - 1, Brown [2] stated the problem "Find the best constant Cn such that p '(z) does not vanish in |z| < Cn". He also conjectured in the same paper that Cn = 1n. This problem was solved by Aziz and Zarger [1]. In this paper, we obtain the results which generalizes the results of Aziz and Zarger.