Froissart bound and self-similarity based models of proton structure functions

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 $1/x$.