The Froissart bound for inelastic cross-sections

We prove that while the total cross{}-section is bounded by $(\pi/m_\pi^2) \ln^2 s$,where$s$isthesquareofthec\mathrm{.}m\mathrm{.}energyand$m_\pi$ the mass of the pion, the total inelastic cross{}-section is bounded by $(1/4)(\pi/m_\pi^2) \ln^2 s$, which is 4 times smaller. We discuss the implications of this result on the total cross{}-section itself.