Bounds for the zeros of polynomials from matrix inequalities

Abstract

We apply several matrix inequalities to the Frobenius companion matrices of monic polynomials to derive new bounds for the zeros of these polynomials. Our analysis enables us to improve an earlier bound of Abdurakhmanov and to give a different proof of a known bound of Fujii and Kubo.

Numerical Radius Inequalities for Certain 2 × 2 Operator Matrices

“…For a useful numerical radius compression inequality, together with an improvement and applications, we refer to [2,6], and [8].…”

Bounds and a majorization for the real parts of the zeros of polynomials

“…By considering the imaginary part of C(p), an analysis similar to that used in deriving the bounds (12) enables us to obtain analogous bounds for the imaginary parts of the zeros of p. These bounds, when combined with (12), describe a rectangle that contains all the zeros of p. Related rectangles have been described in [11,Theorem 3].…”

“…See, e.g., [7, p. 316]. Using a numerical radius estimation of C(p), it has been shown in [5] (see also [11] for a different proof) that if z is any zero of p, then…”

Bounds and Majorization Relations for the Zeros of Polynomials

“…Estimates for the spectral radii of C p , C 2 p , and C 3 p , and the numerical radii of C p and C 2 p have been used to give bounds for the zeros of p (see, e.g., [1,2,8,9,[11][12][13]). In this section, we use a recent numerical radius inequality obtained in [1] to establish a new bound for the zeros of p. This inequality improves an earlier inequality of Hou and Du [5] concerning the numerical radii of partitioned matrices.…”