  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.
