The behavior of the heavy-quarks structure functions at small-$x$

The behavior of the charm and bottom structure functions ($F_k^i(x,Q^2)$, i=c,b; k=2,L) at small-$x$ is considered with respect to the hard-Pomeron and saturation models. Having checked that this behavior predicate the heavy flavor reduced cross sections concerning the unshadowed and shadowed corrections. We will show that the effective exponents for the unshadowed and saturation corrections are independent of $x$ and $Q^2$, and also the effective coefficients are dependent to $\ln Q^2$ compared to Donnachie-Landshoff (DL) and color dipole (CD) models.