Longitudinal structure function from logarithmic slopes of F2 at low x

Using Laplace transform techniques, I calculate the longitudinal structure function F-L (x, Q(2)) from the scaling violations of the proton structure function F-2(x, Q(2)) and make a critical study of this relationship between the structure functions at leading order (LO) up to next-to-next-to leading order (NNLO) analysis at small x. Furthermore, I consider heavy quark contributions to the relation between the structure functions, which leads to compact formula for N-f = 3 + Heavy. The nonlinear corrections to the longitudinal structure function at LO up to NNLO analysis are shown in the N-f = 4 (light quark flavor) based on the nonlinear corrections at R = 2 and R = 4 GeV-1. The results are compared with experimental data of the longitudinal proton structure function FL in the range of 6.5 <= Q(2) <= 800 GeV2.