On annuli containing all the zeros of a polynomial

Abstract

a b s t r a c tIn this paper, we obtain the annuli that contain all the zeros of the polynomial p(z) = a 0 + a 1 z + a 2 z 2 + · · · + a n z n , where a i 's are complex coefficients and z is a complex variable. Our results sharpen some of the recently obtained results in this direction. Also, we develop a MATLAB code to show that for some polynomials the bounds obtained by our results are considerably sharper than the bounds obtainable from the known results.

Eneström–Kakeya Theorem and Some of Its Generalizations

“…Govil and Rahman [30] also gave this type of result, and the same is stated as follows: Related results concerning the location of the zeros of a polynomial have also been presented by Aziz and Mohammad [7], Sun and Hsieh [61], Affane-Aji, Agarwal, and Govil [2], Affane-Aji, Biaz and Govil [3], Choo [15], Choo and Choi [17], Dalal and Govil [19], Gulzar [34,36], and Gilani [27]. The hypotheses of the following result, due to Jain [43] in 1988, are very much in the spirit of the Eneström-Kakeya theorem, although the conclusion involves the size of the real part of the zeros instead of the modulus: In the same paper, Jain gave a result by putting the monotonicity hypothesis on the real parts of the coefficients.…”

“…See [1,2,3,19] for several related results which apply to all polynomials with complex coefficients. In this survey, we explore the Eneström-Kakeya theorem and its generalizations.…”

Bound for the zeros of polynomials

“…Also Affane, Biaz and Govil [1] have obtained a lower bound for the zeros of polynomials in terms of binomial coefficients by proving:…”

A Note on Comparison of Annuli Containing all the Zeros of a Polynomial

“…The above result of Cauchy has been sharpened among others by Joyal et al [11], Datt and Govil [8], Affane-Aji et al [1], and Sun and Hsieh [16].…”