On the Eneström-Kakeya theorem and its sharpness

Abstract A new proof, based on the Perron-Frobenius theory of nonnegative matrices, is given of a result of Hurwitz on the sharpness of the classical Enestrom-Kakeya theorem for estimating the moduli of the zeros of a polynomial with positive real coefficients. It is then shown (Theorem 2) that the zeros of a particular set of polynomials fill out the Enestrom-Kakeya annulus in a precise manner, and this is illustrated by numerical results in Fig. 1.