A derivation is given in terms of the eikonal formalism of a recently derived upper bound on the absorptive part of a spin-zero elastic scattering amplitude. The multivaluedness inherent in the problem is explored, and considerably smaller upper bounds are obtained by imposing conditions of reasonability on the profile function. Experimental differential cross sections for ${\pi }^{\pm }p$ and $\mathrm{pp}$ elastic scattering over a wide energy range are found to fall on a universal curve, which lies just below our most stringent upper bound over a range of three decades in $\frac{d\sigma }{\mathrm{dt}}$.

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