It is shown that if the analytically continued partial-wave amplitude is assumed to have $l$ dependence ${a}_{\ifmmode\pm\else\textpm\fi{}}(s, l)=\ensuremath{\Sigma}\stackrel{n}{l=0}{{C}_{m}}^{\ifmmode\pm\else\textpm\fi{}}(s){l}^{m}{(l+1)}^{m}$ for $l<{l}_{0}(s)$ and finite $n$, the scattering amplitude is bounded by $\mathrm{exp}{\ensuremath{-}\mathrm{const}{[{l}_{0}(s)sin\ensuremath{\theta}(s)]}^{\frac{1}{2}}}$ at high energies. Here ${a}_{+}(s, l)[{a}_{\ensuremath{-}}(s, l)]$ is equal to ${a}_{l}(s)$ for even (odd) integer $l$. The most physical example of this dependence is that in which a central area of the scatterer becomes maximally absorptive.
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