Contribution from far singularities in the $\cos\theta$ plane to the scattering amplitude and to the distribution function of inclusive processes

Using the imitarity condition, polynomial boundedness with respect to energy and analyticity of the amplitude $F(s,z)$ in the $z=\cos\theta$-plane in a certain fixed complex neighbourhood of the physical points $-1<z<1$, it is shown that if the high-energy asymptotics of the amplitude is such that $|F(s,1)|\geqslant c(\ln s)^{2+\varepsilon}$, then such behaviour of the amplitude is completely determined by the nearest to the point $z=1$ singularity of the amplitude. The similar results are obtained for the spectrum of one-particle inclusive process integrated over the momentum values. It is also shown that if the absorptive part of the elastic scattering amplitude is analytic in a certain bounded region of the $z$-plane with cuts along the real axis and $\sigma_{\mathrm {tot}}(s)>(\ln s)^{-1}$ then the discontinuity of the amplitude on the right-hand side cut is a sign-changing function of $z$.