  Proof of the Froissart theorem is reconsidered in a different way to extract
its necessary conditions. Two physical inputs, unitarity and absence of
massless intermediate hadrons, are indisputable. Also important are
mathematical properties of the Legendre functions. Assumptions on dispersion
relations, single or double, appear to be excessive. Instead, one should make
assumptions on possible high-energy asymptotics of the amplitude in nonphysical
configurations, which have today no firm basis. Asymptotics for the physical
amplitude always appear essentially softer than for the nonphysical one.
Froissart's paper explicitly assumed the hypothesis of power behavior and
obtained asymptotic bound for total cross sections $\sim \log^2{(s/s_0)}$ with
some constant $s_0$. Our bounds are slightly stronger than original Froissart
ones. They show that the scale $s_0$ should itself slowly grow with $s$. Under
different assumptions about asymptotic behavior of nonphysical amplitudes, the
total cross section could grow even faster than $\log^2{s}$. The problem of
correct asymptotics might be clarified by precise measurements at the LHC and
higher energies.
