Sum rules for the ratio of the $\pi^{\pm} p$-scattering amplitudes

Analyticity, unitarity, and crossing symmetry are used to obtain an exact integral relationship between $|f_{+}(E)/f_{-}(E)|$ and the difference of the phases of $f_{+}(E)$ and $f_{-}(E)$ and the analogous relationship between $|f_{+}(E)f_{-}(E)|$ and the sum of the phases of these amplitudes [$(f_{\pm}(E)$ are the $\pi^{\pm}-p$-forward scattering amplitudes]. Restrictions on $\displaystyle\int_{1}^{E}\ln\biggl|\frac{f_{+}(E')}{f_{-}(E')}\biggr|\, \frac{dE'}{\sqrt{{E'}^2-1}}$ are found.