The method of Singh and Roy, which obtains a bound on the imaginary part of the elastic scattering amplitude, given ${\sigma }_{\mathrm{tot}}$ and ${\sigma }_{\mathrm{el}}$, is generalized to allow inclusion of other given data such as $\frac{d\sigma }{\mathrm{dt}}$ at fixed points. A calculation is described in which one additional experimental point is given. The upper bound is compared with the experimental data for ${\pi }^{\pm }p$, $K^{\pm }p$, $\mathrm{pp}$, and $p\overline{p}$ elastic scattering at various energies in the nearforward direction. It is found that this single extra piece of given data extends by a factor of about 3 the range for which the original bound remained close to the data and at the largest $t$ value considered ($t\approx -0.5\mathrm{}$ Be${\mathrm{V}}^2$) improves this bound by a factor of about 4.

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