On the Rate of Overconvergence of the Generalized Eneström-Kakeya Functional for Polynomials

Abstract

The classical Eneström-Kakeya Theorem, which provides an upper bound for the moduli of zeros of any polynomial with positive coefficients, has been recently extended by Anderson, Saff and Varga to the case of any complex polynomial having no zeros on the ray [0,$+∞$). Their extension is sharp in the sense that, given such a complex polynomials $p_n(z)$ of degree $n≥1$, a sequence of multiplier polynomial can be found for which the Eneström-Kakeya upper bound, applied to the products $Q_{mi}(z)$ · $p_n(z)$, converges, in the limit as $i$ tends to $∞$, to the maximum of the moduli of the zeros of $p_n(z)$. Here, the rate of convergence of these upper bounds is studied. It is shown that the obtained rate of convergence is best possible.