Bounds for diluted mean-fields spin glass models

In an important recent paper, \cite{FL}, S. Franz and M. Leone prove rigorous lower bounds for the free energy of the diluted $p$-spin model and the $K$-sat model at any temperature. We show that the results for these two models are consequences of a single general principle. Our calculations are significantly simpler than those of \cite{FL}, even in the replica-symmetric case.