Abstract
It is shown that if A(r) does not converge to the zero operator on a complex Hilbert space for some r >= 1, then
w(A) <= 1/2 (parallel to A(r)parallel to(1/r) + parallel to A(2)parallel to(1/2)).
And if there exists r such that A(r) is a zero operator on a complex Hilbert space for all r >= 2, then
w(A) = 1/2 parallel to A parallel to,
where w(.) and parallel to.parallel to are the numerical radius and the usual operator norm, respectively. Also, the previous inequalities are better than any other classical inequalities.