Numerical Radius Inequalities Involving Accretive Operators

We investigate some new numerical radius inequalities for accretive operators. More precisely, we prove that if A; A(2) is an element of B(H)\{0}, 0 < m < M, and the operator C-m,C-M(A(2)) = (A* - mI) (MI -A) is accretive (i.e. Re < C-m,C-M (A(2))x,x > >= 0 for all x is an element of H), then 1/2 (parallel to A(2)parallel to(1/2) + parallel to A parallel to) <= (0.0708(m + M/2 root mM)(1/2) + 1)w(A), which is a reverse of the inequality w(A) <= 1/2 (parallel to A(2)parallel to(1/2) + parallel to A parallel to), and sharper than inequality parallel to A parallel to <= 2w(A), under a special condition. Here, w(.) and parallel to.parallel to are the numerical radius and the usual operator norm, respectively.