  Recently Andr\'e Martin has proved a rigorous upper bound on the inelastic
cross-section $\sigma_{inel}$ at high energy which is one-fourth of the known
Froissart-Martin-Lukaszuk upper bound on $\sigma_{tot}$. Here we obtain an
upper bound on $\sigma_{inel}$ in terms of $\sigma_{tot}$ and show that the
Martin bound on $\sigma_{inel}$ is improved significantly with this added
information.
