New inequalities for the A-numerical radius of the products and sums of operators acting on a semi-Hilbert space, i.e. a space generated by a positive semidefinite operator A, are established. In particular, for every operators T and S which admit A-adjoints, it is proved that ω A (TS) ≤ 1 2 ω A (ST) + 1 4 ∥T∥ A ∥S∥ A + ∥TS∥ A , where ω A (T) and ∥T∥ A denote the A-numerical radius and the A-operator seminorm of an operator T respectively