A Lower Bound on Inelasticity in Pion-Pion Scattering

Assuming that the pion-pion scattering amplitude and its absorptive part are analytic inside an ellipse in $t$- plane with foci $t=0$, $u=0$ and right extremity $t=4m_{\pi }^2+ϵ$, ($ϵ>0$), except for cuts prescribed by Mandelstam representation for $t\ge 4m_{\pi }^2$, $u\geq 4 m_{\pi}^2$,andboundedby$s^N$ on the boundary of this domain, we prove that for $s\to \mathrm{\infty }$, \sigma_{inel} (s) > \frac{Const}{s^{5/2} }\exp {[-\frac{\sqrt{s}}{4} (N+5/2) \ln {s} ]}.