  We present some new upper and lower bounds for the numerical radius of
bounded linear operators on a complex Hilbert space and show that these are
stronger than the existing ones. In particular, we prove that if $A$ is a
bounded linear operator on a complex Hilbert space $\mathcal{H}$ and if
$\Re(A)$, $\Im(A)$ are the real part, the imaginary part of $A$, respectively,
then $$ w(A)\geq\frac{\|A\|}{2} +\frac{1}{2\sqrt{2}} \Big |
\|\Re(A)+\Im(A)\|-\|\Re(A)-\Im(A)\| \Big | $$ and $$
w^2(A)\geq\frac{1}{4}\|A^*A+AA^*\|+\frac{1}{4}\Big|
\|\Re(A)+\Im(A)\|^2-\|\Re(A)-\Im(A)\|^2\Big|. $$ Here $w(.)$ and $\|.\|$ denote
the numerical radius and the operator norm, respectively. Further, we obtain
refinement of inequalities for the numerical radius of the product of two
operators. Finally, as an application of the second inequality mentioned above,
we obtain an improvement of upper bound for the numerical radius of the
commutators of operators.
