If A, B are bounded linear operators on a complex Hilbert space, then we prove that w(A)≤12(‖A‖+r(|A||A∗|)),w(AB±BA)≤22‖B‖w2(A)-c2(R(A))+c2(I(A))2,where w(·) , ∥ · ∥ , and r(·) are the numerical radius, the operator norm, the Crawford number, and the spectral radius respectively, and R(A) , I(A) are the real part, the imaginary part of A respectively. The inequalities obtained here generalize and improve on the existing well known inequalities.