Unitarity bound on the total cross section and forward slope of the deep inelastic $\pi N$ scattering amplitude

The generalized method of Lagrangian multipliers is used to solve in explicit analytic form the problem of unitarity maximization of the total cross section $\sigma_t$ of deep inelastic scattering of particles with spin $0$ on spin $1/2$ particles at all energies. The upper bound $\sigma_t^{\max}$ ax is expressed in terms of the total elastic cross sectione $\sigma_e$ and the forward slope $b_+$ of the imaginary part of the helicity nonflip amplitude. A lower bound is also found for $b_+$ in terms of $\sigma_e$ and $\sigma_t$, this making more precise the well-known bound of MacDowell and Martin for the case of meson-nucleon scattering.