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

TL;DR

This paper develops improved bounds for the zeros of monic polynomials by leveraging numerical radius estimates of 2x2 operator matrices, generalizing previous inequalities and demonstrating superior bounds through numerical examples.

Contribution

It introduces new bounds for the numerical radius of operator matrices that generalize and improve existing inequalities, leading to better estimates for polynomial zeros.

Findings

New bounds for the numerical radius of 2x2 operator matrices

Generalization of existing inequalities for numerical radius

Numerical examples showing improved bounds for polynomial zeros

Abstract

We obtain bounds for the numerical radius of $2 \times 2$ 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.