Regions of analyticity in a model three-component solution

A model for a three-component solution is considered in which the bonds of a square or cubic lattice are covered by molecules of types $\mathrm{AA}$, $\mathrm{BB}$, and $\mathrm{AB}$. The ends of molecules near a common lattice site interact with energies ${\epsilon }_{\mathrm{AA}}$, ${\epsilon }_{\mathrm{BB}}$, and ${\epsilon }_{\mathrm{AB}}$. It is shown there is no demixing transition in the model if ${\epsilon }_{\mathrm{AB}}<~\frac{({\epsilon }_{\mathrm{AA}}+{\epsilon }_{\mathrm{BB}})}{2}$.