Singularities of the amplitude in the $\cos\theta$; plane and scattering at high energies

Unitarity and analyticity of the elastic scattering amplitude in the $\cos\theta$ plane are used to show that if $d\sigma/dt|_{t=0}\geqslant\text{const}/s^{2-\varepsilon}$ $(\varepsilon>0)$ at high energies then the contribution to the differential cross section of forward elastic scattering from the left-hand "$u$-channel" singularities of the amplitude is negligibly small.