Annular bounds for the zeros of a polynomial from companion matrix

Abstract

Let $p(z)=z^n+a_{n-1}{z}^{n-1}+a_{n-2}{z}^{n-2}+\dots +a_{1}z+a_0$ be a complex polynomial with $a_0\ne 0$ and $n\ge 3$. Several new upper bounds for the moduli of the zeros of $p$ are developed. In particular, if $\alpha =\sqrt{{\sum }_{j=0}^{n-1}|a_j{|}^2}$ and $z$ is any zero of $p$, then we show that \begin{eqnarray*} |z|^2 &\leq & \cos^2 \fracπ{n+1}+|a_{n-2}|+ \frac{1}{4} \left ( |a_{n-1}|+ { α} \right)^2 + \frac{1}{2}\sqrt{α^2-|a_{n-1}|^2} + \frac{1}{2}α, \end{eqnarray*} which is sharper than the Abu-Omar and Kittaneh's bound \begin{eqnarray*} |z|^2 &\leq & \cos^2 \fracπ{n+1}+ \frac{1}{4} \left ( |a_{n-1}|+ { α}\right)^2 + α \end{eqnarray*} if and only if $2|a_{n-2}|<\sqrt{{\sum }_{j=0}^{n-1}|a_j{|}^2}-\sqrt{{\sum }_{j=0}^{n-2}|a_j{|}^2}.$ The upper bounds obtained here enable us to describe smaller annuli in the complex plane containing all the zeros of $p$.

References in corpus (1)

• Estimations of zeros of a polynomial using numerical radius inequalities

Cited by in corpus (1)

• Numerical radius inequalities and estimation of zeros of polynomials