  If $A,B$ are bounded linear operators on a complex Hilbert space, then %
$w(A) \leq \frac{1}{2}\left( \|A\|+\sqrt{r\left(|A||A^*|\right)}\right)$ and
$w(AB \pm BA)\leq 2\sqrt{2}\|B\|\sqrt{ w^2(A)-\frac{c^2(\Re (A))+c^2(\Im
(A))}{2} },$ \begin{eqnarray*} w(A) &\leq& \frac{1}{2}\left(
\|A\|+\sqrt{r\left(|A||A^*|\right)}\right),\\ w(AB \pm BA)&\leq&
2\sqrt{2}\|B\|\sqrt{ w^2(A)-\frac{c^2(\Re (A))+c^2(\Im (A))}{2} },
\end{eqnarray*} where $w(.),\|.\|,c(.)$ and $r(.)$ are the numerical radius,
the operator norm, the Crawford number and the spectral radius respectively,
and $\Re (A)$, $\Im (A)$ are the real part, the imaginary part of $A$
respectively. The inequalities obtained here generalize and improve on the
existing well known inequalities.
