  We develop upper and lower bounds for the numerical radius of $2\times 2$
off-diagonal operator matrices, which generalize and improve on the existing
ones. We also show that if $A$ is a bounded linear operator on a complex
Hilbert space and $|A|$ stands for the positive square root of $A$, i.e.,
$|A|=(A^*A)^{1/2}$, then for all $r\geq 1$, $w^{2r}(A) \leq \frac{1}{4} \big \|
|A|^{2r}+|A^*|^{2r} \big \| + \frac{1}{2} \min\left\{ \big \|\Re\big(|A|^r\,
|A^*|^r \big) \big \|, w^r(A^2) \right\} $ where $w(A)$, $\|A\|$ and $\Re(A)$,
respectively, stand for the numerical radius, the operator norm and the real
part of $A$. This (for $r=1$) improves on existing well-known numerical radius
inequalities.
