A lower bound on the Longitudinal Structure Function at small x from a self-similarity based model of Proton

Self-similarity based model of proton structure function at small \textit{x} was reported in the literature sometime back. The phenomenological validity of the model is in the kinematical region $6.2\, \times \, 10^{-7} \leq x \leq 10^{-2}$ and $0.045 \leq Q^{2} \leq 120 \, \mathrm{GeV^{2}}$. We use momentum sum rule to pin down the corresponding self-similarity based gluon distribution function valid in the same kinematical region. The model is then used to compute bound on the longitudinal structure function $F_{L}\left(x,Q^{2} \right)$ for Altarelli-Martinelli equation in QCD and is compared with the recent HERA data.