Sharp Inequalities for the Numerical Radii of Block Operator Matrices

In this paper we present several sharp upper bounds for the numerical radii of the diagonal and off-diagonal parts of the 2 x 2 block operator matrix [(A)(C) (B)(D)]. Among extensions of some results of Kittaneh et al., it is shown that if T = [(A)(0) (0)(D)], and f and g are non-negative continuous functions on [0, infinity) such that f(t)g(t) = t (t >= 0), then for all non-negative nondecreasing convex functions h on [0, infinity), we obtain that h(w(r)(T)) <= max (parallel to 1/p h(f(pr) (vertical bar A vertical bar)) + 1/q h(g(qr) (vertical bar A*vertical bar))parallel to,parallel to 1/p h(f(pr)(vertical bar D vertical bar)) + 1/q h(g(qr) (vertical bar D*vertical bar))parallel to), where p, q > 1 with 1/p + 1/q = 1, and r min(p, q) >= 2.