JOURNAL Ot APPROXIMATION THEORY 22, l-10 (1978) On the Enestrijm-Kakeya Theorem, II N. K. COWL Department of Mathematics, I.I.T., New Delhi-110029, India AND V. K. JAIN Department of Mathematics, I.1. T,, Kharagpur-721302, India Communicated by Oued Shisha Received December 18, 1975 1. INTRODUCTION AND STATEMENT OF RESULTS The following result is well known in the theory of the distribution of zeros of polynomials. THEOREM A (Enestriim-Kakeya). If a, 3 a,-, > an-2 > ... > a, 3 a, ) 0, then, for I z / > 1, Cg, akzk # 0. (1.1) There already exist in the literature [l; 3, Theorems 1-4; 5, Theorem 3; 63 some extensions of the Enestrom-Kakeya theorem. Govil and Rahman [3, Theorems 2, 41 generalized this theorem to polynomials with complex coefficients, first by considering the moduli of the coefficients to be monoto- nically increasing and then by assuming the real parts of the coefficients to be monotonically increasing. While refining the results of Govil and Rahman [3, Theorems 2, 41, we [2] proved the following THEOREM B. Let p(z) = Ci=,, aKzlC be a polynomial with complex coeffi- cients such that I arg 4 - P I < a: < 74, k = 0, l,..., n, for some real ,0, and I a, I 2 I a,-, I 3 a.* > I aI I 2 I a, I > 0; 1 (1.2) 0021-9045/78/0221-0001$02.00/0 Copyright 8 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.