In this paper we present certain interesting refinements of a well-known EnestromKakeya theorem in the theory of distribution of zeros of polynomials which among other things also improve upon some results of Aziz and Mohammad, Govil and Rehman and others.

In this paper we shall obtain some interesting extensions and generalizations of a well-known theorem due to Enestrom and Kakeya according to which all the zeros of a polynomial  

Let p be a complex polynomial, of the form , where |zk| ≥ 1 when 1 ≤ k ≤ n − 1. Then p′(z) ≠ 0 if |z| /n.Let B(z, r) denote the open ball in with centre z and radius r, and denote its closure. The Gauss-Lucas theorem states that every critical point of a complex polynomial p of degree at least 2 lies in the convex hull of its zeros. This theorem has been further investigated and developed. B. Sendov conjectured that, if all the zeros of p lie in then, for any zero ζ of p, the disc contains at least one zero of p′; see [3, Problem 4.1]. This conjecture has attracted much attention-see, for example, [1], and the papers cited there. In connection with this conjecture, Brown [2] posed the following problem.

Abstract. Let P(z) be a polynomial of degree n and P (z) its derivative. In this paper we shall obtain certain sharp generalization of some results of Govil, Malik and P. Turān concerning the maximum modulus of P(z) and P (z) .Mathematics subject classification (1991): 30A06, 30A64.

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