  Assuming that axiomatic local field theory results hold for hadron
scattering, Andr\'e Martin and S. M. Roy recently obtained absolute bounds on
the D-wave below threshold for pion-pion scattering and thereby determined the
scale of the logarithm in the Froissart bound on total cross sections in terms
of pion mass only. Previously, Martin proved a rigorous upper bound on the
inelastic cross-section $\sigma_{inel}$ which is one-fourth of the
corresponding upper bound on $\sigma_{tot}$, and Wu, Martin,Roy and Singh
improved the bound by adding the constraint of a given $\sigma_{tot}$. Here we
use unitarity and analyticity to determine, without any high energy
approximation, upper bounds on energy averaged inelastic cross sections in
terms of low energy data in the crossed channel. These are Froissart-type
bounds without any unknown coefficient or unknown scale factors and can be
tested experimentally. Alternatively, their asymptotic forms,together with the
Martin-Roy absolute bounds on pion-pion D-waves below threshold, yield absolute
bounds on energy-averaged inelastic cross sections. E.g. for $\pi^0 \pi^0$
scattering, defining $\sigma_{inel}=\sigma_{tot} -\big (\sigma^{\pi^0 \pi^0
\rightarrow \pi^0 \pi^0} + \sigma^{\pi^0 \pi^0 \rightarrow \pi^+ \pi^-} \big
)$,we show that for c.m. energy $\sqrt{s}\rightarrow \infty $,
$\bar{\sigma}_{inel }(s,\infty)\equiv s\int_{s} ^{\infty } ds'\sigma_{inel
}(s')/s'^2 \leq (\pi /4) (m_{\pi })^{-2} [\ln (s/s_1)+(1/2)\ln \ln (s/s_1)
+1]^2$ where $1/s_1= 34\pi \sqrt{2\pi }\>m_{\pi }^{-2} $ . This bound is
asymptotically one-fourth of the corresponding Martin-Roy bound on the total
cross section, and the scale factor $s_1$ is one-fourth of the scale factor in
the total cross section bound. The average over the interval (s,2s) of the
inelastic $\pi^0 \pi^0 $cross section has a bound of the same form with $1/s_1$
replaced by $1/s_2=2/s_1 $.
