A proof of the existence of functions that satisfy exactly both crossing and unitarity IV. nearly constant asymptotic cross sections

A proof of the existence of functions that satisfy a Mandelstam representation with the crossing symmetry appropriate to pion-pion scattering, elastic unitarity below the inelastic threshold, and the inelastic inequalities above it, is generalized to a restricted class of amplitudes that need an arbitrary, finite number of subtractions. Although the leading angular momentum singularity is constrained not to rise above unity in the elastic region, it is possible for the total cross section to display an almost constant asymptotic behaviour.