  Froissart Bound implies that the total proton-proton cross-section (or
equivalently structure function) cannot rise faster than the logarithmic growth
$\log^2 s \sim \log^2 1/x$, where \textit{s} is the square of the center of
mass energy and \textit{x} is the Bjorken variable. Compatibility of such
behavior with the notion of self-similarity in a model of structure function
suggested by us sometime back is now generalized to more recent improved
self-similarity based models and compare with recent data as well as with the
model of Block, Durand, Ha and McKay. Our analysis suggests that Froissart
bound compatible self-similarity based models are possible with $\log^2 1/x$
rise in limited $x-Q^2$ ranges of HERA data, but their phenomenological ranges
validity are narrower than the corresponding models having power law rise in
$1/x$.
