  A classical result by Buchdahl \cite{Bu1} shows that for static solutions of
the spherically symmetric Einstein-matter system, the total ADM mass M and the
area radius R of the boundary of the body, obey the inequality $2M/R\leq 8/9.$
The proof of this inequality rests on the hypotheses that the energy density is
non-increasing outwards and that the pressure is isotropic. In this work
neither of Buchdahl's hypotheses are assumed. We consider non-isotropic
spherically symmetric shells, supported in $[R_0,R_1], R_0>0,$ of matter models
for which the energy density $\rho\geq 0,$ and the radial- and tangential
pressures $p\geq 0$ and $q,$ satisfy $p+q\leq\Omega\rho, \Omega\geq 1.$ We show
a Buchdahl type inequality for shells which are thin; given an $\epsilon<1/4$
there is a $\kappa>0$ such that $2M/R_1\leq 1-\kappa$ when $R_1/R_0\leq
1+\epsilon.$ It is also shown that for a sequence of solutions such that
$R_1/R_0\to 1,$ the limit supremum of $2M/R_1$ of the sequence is bounded by
$((2\Omega+1)^2-1)/(2\Omega+1)^2.$ In particular if $\Omega=1,$ which is the
case for Vlasov matter, the boumd is $8/9.$ The latter result is motivated by
numerical simulations \cite{AR2} which indicate that for non-isotropic shells
of Vlasov matter $2M/R_1\leq 8/9,$ and moreover, that the value 8/9 is
approached for shells with $R_1/R_0\to 1$. In \cite{An2} a sequence of shells
of Vlasov matter is constructed with the properties that $R_1/R_0\to 1,$ and
that $2M/R_1$ equals 8/9 in the limit. We emphasize that in the present paper
no field equations for the matter are used, whereas in \cite{An2} the Vlasov
equation is important.
