  We derive generalisations of the Weingarten--Witten QCD mass inequalities for
particular multi-hadron systems. For systems of any number of identical
pseudo-scalar mesons of maximal isospin, these inequalities prove that
interactions between the constituent mesons must be repulsive and that no bound
states can form in these channels. Similar constraints in less symmetric
systems are also extracted. These results are compatible with experimental
results (where known) and recent lattice QCD calculations, and also lead to a
more stringent bound on the nucleon mass than previously derived, $m_N \ge 3/2
m_\pi$.
