Bounds for zeros of a polynomial using numerical radius of Hilbert space operators

Abstract

We obtain bounds for the numerical radius of 2 × 2 operator matrices which improve on the existing bounds. We also show that the inequalities obtained here generalize the existing ones. As an application of the results obtained here, we estimate the bounds for the zeros of a monic polynomial and illustrate with numerical examples that the bounds are better than the existing ones.

Improved inequalities for the numerical radius via Cartesian decomposition

“…Thus (i) holds. Also, from (i) of Remark 2.4, we have ℜ(A) + ℑ(A) 2 = ℜ(A) − ℑ(A) 2 . In addition, we conclude from Theorem 2.9 that ℜ(A) + ℑ(A) 2 = 1 2 A * A + AA * , which yields (ii).…”

“…The first inequality becomes equality if A 2 = 0, and the second one turns into equality if A is normal. Over the years, many mathematicians have obtained various refinements of (1.1), we refer the reader to [1,2,9,12,13,14] and references therein. In particular, Kittaneh [10] improved the inequalities in (1.1) by establishing that…”

“…A) + ℑ(A)2 , ℜ(A) − ℑ(A) 2 + ℜ(A) + ℑ(A) ℜ(A) − ℑ(A) obtain an upper bound for the numerical radius of bounded linear operators.The following inequality is known as the Buzano inequality.Lemma 2.15. ([6]) Let x, y, e ∈ H with e = 1.…”

Improvement of numerical radius inequalities

“…The first inequality becomes equality if A 2 = O, and the second one turns into equality if A is normal. For various refinements of (1.1), we refer the reader to [3,4,5,6,9,17] and references therein. In particular, Kittaneh [13] improved the inequalities in (1.1) by establishing that…”

Annular bounds for the zeros of a polynomial from companion matrix

“…The Frobenius companion matrix plays an important link between matrix theory and the geometry of polynomials. It has been used to obtain estimations for zeros of polynomials by matrix methods, we refer to some of the recent papers [3,4,5,13] and the references therein. Also, various mathematicians have obtained annular regions containing all the zeros of a polynomial by using classical approach, we refer to [6,7,15] and references therein.…”