Finite energy bounds for $\piN$ scattering

Upper bounds on energy averaged pi N cross sections are given. Using low energy data and data from pi N backward scattering and NN to pi pi annihilation, it is found that sigma /sub tot/<or approximately=80 mb and sigma /sub inel/<or approximately=30 mb, for energies just above the phase-shift region. The bounds are based on assumptions similar to those underlying Froissart's bound and are equal to it asymptotically. However, at finite but large energies, they increase much slower than what might have been anticipated on purely numerological grounds. Related problems in pp and Kp scattering are also discussed. (25 refs) .