On the Robustness property of Eneström-Kakeya theorem

Abstract

The Eneström-Kakeya theorem states that an nth-order polynomial P(z) = ∑ n i=0 a iz i with positive coefficients has all its zeros in the disk |z| ≤ 1 if its coefficients monotonically decrease, i.e., a n ≥ a n-1 ≥...≥ a 1 ≥ a 0 > 0. In the literature some attempts have been made to extend and generalize the Eneström-Kakeya theorem. In this paper we study the robustness property of the Eneström-Kakeya theorem. It is shown that even if the monotonicity is violated by one coefficient (say a k), then all the zeros of P(z) still remain in the disk |z| ≤ 1 if the deviation of ak from a k+1 or a k-1 is not too large. More precisely we derive the upper bounds for the deviations of a k from a k+1 or a k-1 to ensure that P(z) has all its zeros in the disk |z| ≤ 1.