The Longitudinal Structure Function F$_L$ as a Function of F$_2$ and   dF$_2$/dlnQ$^2$ at small x. The Next-to-Leading Analysis
  We present a set of formulae to extract the longitudinal deep inelastic structure function $F_L$ from the transverse structure function $F_2$ and its derivative $dF_2/dlnQ^2$ at small $x$. Our expressions are valid for any value of $\delta$, being $x^{-\delta}$ the behavior of the parton densities at low $x$. Using $F_2$ HERA data we obtain $F_L$ in the range $10^{-4} \leq x \leq 10^{-2}$ at $Q^2=20$ GeV$^2$. Some other applications of the formulae are pointed out. 
A.V. Kotikov and G. Parente