On the zeros of a polynomial and its derivative

then every zero of the derivative P'(z) is contained in the smallest convex set that contains 8. This theorem has been rather thoroughly investigated [2] and sharpened in several ways. However, there is one related question that deserves attention, namely given one specific zero z,b of P(z), what can be said about a neighborhood of zX that will always contain a zero of P'(z)? By a translation, followed by a stretching (or shrinking), we may require that all the zeros of P(z) are in 8, the closed unit disk. Further by a rotation we may set zS = a, where 0?< a ? 1. With this normalization the problem can now be formulated precisely. Let 6P(a, n) be the set of all nth degree polynomials P(z) that have all of their zeros in 8, and at least one zero at z = a, 0? a <1. Let D(a, n) be a minimal region with the property that if P(z) zP(a, n), then P'(z) has at least one zero in 5O(a, n). Describe 5O(a, n). This problem seems to be rather difficult. As far as the authors are aware, the first step in the direction of this problem is contained in the conjecture due to Iliev [1, p. 25]. Let g(a) be the intersection of the disk j z-a-aJ ?1 with 8. Then, according to lliev, g(a) always contains at least one zero of P'(z). In this paper we replace g(a) by a region C*(a) which is in general much smaller, and we conjecture that every P(z) in 6'(a, n), has at least one root in C*(a). We can prove this proposition if a= 1. The case a = 0 is trivial. For 0 a < 1, the conjecture is still open.