  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 $p$.
