  The Froissart bound implies that the total cross section (or, equivalently,
the structure function) cannot rise faster than the logarithmic growth of $
\ln^{2} \left(\frac{1}{x} \right)$. In this work, we show that such a slow
growth is not compatible with the notion of self-similarity. As a result, it
calls for the modification of the defining transverse-momentum-dependent parton
density function (TMD PDF) of a self-similarity based proton structure function
$F_{2} \left(x,Q^{2} \right)$ at small \textit{x}. Using plausible assumptions,
we obtain the Froissart saturation condition on this TMD PDF.
