It is shown that if A is a bounded linear operator on a complex Hilbert space, then
1/4 ||A*A + AA*|| ≤ (w(A))² ≤ 1/2 ||A*A + AA*||,
where w(·) and ||·|| are the numerical radius and the usual operator norm, respectively. These inequalities lead to a considerable improvement of the well known inequalities
1/2 ||A|| ≤ w(A) ≤ || A||.
Numerical radius inequalities for products and commutators of operators are also obtained.