  We prove that while the total cross{}-section is bounded by $(\pi/m_\pi^2)
\ln^2 s$, where $s$ is the square of the c.m. energy and $m_\pi$ 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.
