Improved inequalities related to the A-numerical radius for commutators of operators

Let A be a positive bounded linear operator on a complex Hilbert space H and B-A(H) be the subspace of all operators which admit A-adjoints operators. In this paper, we establish some inequalities involving the commutator and the anticommutator of operators in semi-Hilbert spaces, i.e. spaces generated by positive semidefinite sesquilinear forms. Mainly, among other inequalities, we prove that for T, S is an element of B-A(H) we have omega(A)(TS +/- ST) <= 2 root 2 min {f(A)(T, S), f(A)(S, T)}, where f(A)(X, Y) = parallel to Y parallel to(A) root omega(2)(A) (X) - vertical bar parallel to X+X-#A/2 parallel to(2)(A) - parallel to X-X-#A/2i parallel to(2)(A)vertical bar/2. This covers and improves the well-known inequalities of Fong and Holbrook. Here omega(A)(.) and parallel to.parallel to(A) are the A-numerical radius and the A-operator seminorm of semi-Hilbert space operators, respectively and X-#A denotes a distinguished A-adjoint operator of X.