Sharper bounds for the zeros of polynomials

Abstract

In this paper, new upper bounds for the magnitudes of the zeros of polynomials are developed. These bounds are derived from the Cauchy classical bound applied to a new polynomial having zeros with magnitudes that are powers of those of the original polynomial. Lower and upper bounds for the minimum and maximum zeros of real polynomials with real zeros are also developed. Additionally, we derive Kantorovich like inequalities which are used to derive bounds for the condition number and for the eigen spread of real symmetric matrices. The proposed bounds are tested and compared with many existing bounds using several examples.