Annular bounds for the zeros of a polynomial from companion matrices

Abstract

Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\cdots +a_1z+a_0$ 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

$$\begin{aligned} |z|^2 \le \cos ^2 \frac{\pi }{n+1}+|a_{n-2}|+ \frac{1}{4} \left( |a_{n-1}|+ { \alpha } \right) ^2 + \frac{1}{2}\sqrt{\alpha ^2-|a_{n-1}|^2} + \frac{1}{2}{\alpha }, \end{aligned}$$

which is sharper than the existing bound, given as,

$$\begin{aligned} |z|^2\le & {} \cos ^2 \frac{\pi }{n+1}+ \frac{1}{4} \left( |a_{n-1}|+ { \alpha }\right) ^2 + {\alpha }, \end{aligned}$$

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.