Let G = (V, E) be a simple connected graph of order n ≥ 2, size m with normalized Laplacian eigenvalues γ1 ≥ γ2 ≥ · · · ≥ γn−1 > γn = 0. Denote with (Formula Presented), where α is an arbitrary real number, the sum of powers of normalized Laplacian eigenvalues of graphs. In this paper several inequalities involving invariants of the form sα(G), for various real α are proved. Our results not only generalize and improve some previous results on sα (G), Kemeny constant and Laplacian incidence energy, but also present new bounds for these graph invariants.