Annular bounds for the zeros of a polynomial from companion matrix

Abstract

Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_1z+a_0$ be a complex polynomial with $a_0\neq 0$ and $n\geq 3$. 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 \frac{\pi}{n+1}+|a_{n-2}|+ \frac{1}{4} \left ( |a_{n-1}|+ { \alpha} \right)^2 + \frac{1}{2}\sqrt{\alpha^2-|a_{n-1}|^2} + \frac{1}{2}{\alpha}, \end{eqnarray*} which is sharper than the Abu-Omar and Kittaneh's bound \begin{eqnarray*} |z|^2 &\leq & \cos^2 \frac{\pi}{n+1}+ \frac{1}{4} \left ( |a_{n-1}|+ { \alpha}\right)^2 + {\alpha} \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 $p$.