  In 1959 Buchdahl \cite{Bu} obtained the inequality $2M/R\leq 8/9$ under the
assumptions that the energy density is non-increasing outwards and that the
pressure is isotropic. Here $M$ is the ADM mass and $R$ the area radius of the
boundary of the static body. The assumptions used to derive the Buchdahl
inequality are very restrictive and e.g. neither of them hold in a simple soap
bubble. In this work we remove both of these assumptions and consider
\textit{any} static solution of the spherically symmetric Einstein equations
for which the energy density $\rho\geq 0,$ and the radial- and tangential
pressures $p\geq 0$ and $p_T,$ satisfy $p+2p_T\leq\Omega\rho, \Omega>0,$ and we
show that $$\sup_{r>0}\frac{2m(r)}{r}\leq
\frac{(1+2\Omega)^2-1}{(1+2\Omega)^2},$$ where $m$ is the quasi-local mass, so
that in particular $M=m(R).$ We also show that the inequality is sharp. Note
that when $\Omega=1$ the original bound by Buchdahl is recovered. The
assumptions on the matter model are very general and in particular any model
with $p\geq 0$ which satisfies the dominant energy condition satisfies the
hypotheses with $\Omega=3.$
