The application of the Cerulus-Martin theorem to production amplitudes is considered and a lower bound of the form e −Cs γ logs 1 2 ⩽ γ ⩽ 1 is obtained in a certain subset of the fixed-angle limits of such amplitudes defined in a previous paper. The value of γ obtained depends on the assumptions made which are related to those of multi-Regge theory. It is also shown that the experimentally observed exponential decrease of inclusive differential cross sections in the fixed-angle, fixed scaled longitudinal momentum limit implies a similar decrease for production amplitudes in fixed- angle limits. As a result the lower bound derived for production amplitudes will also apply to inclusive differential cross sections in such a limit.