Improvement of numerical radius inequalities

Let H be a complex Hilbert space with the inner product 〈·, ·〉 and the corresponding norm ‖ · ‖ induced by the inner product. Let B(H ) denote the C-algebra of all bounded linear operators on H with the identity operator I and the zero operator O. Let A ∈ B(H ). We denote by |A| = (AA) 1 2 the positive square root of A, and R(A) = 1 2 (A + A) and I(A) = 1 2i (A − A), respectively, stand for the real and imaginary part of A. The numerical range of A, denoted by W (A), is defined as W (A) = {〈Ax, x〉 : x ∈ H , ‖x‖ = 1} . We denote by ‖A‖ and w(A) the operator norm and the numerical radius of A, respectively, and are defined as