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Given two couplings between "primal" and "dual" sets, we prove a generalimplication that relates an inequality involving "primal" sets to a reverseinequality involving the "dual" sets.% More precisely, let be given two"primal" sets $\PRIMAL$, $\PRIMALBIS$and two "dual" sets $\DUAL$, $\DUALBIS$,together with two {coupling} functions $\PRIMAL\overset{\coupling}{\leftrightarrow} \DUAL $ and $\PRIMALBIS\overset{\couplingbis}{\leftrightarrow} \DUALBIS $. We define a new coupling$\SumCoupling{\coupling}{\couplingbis} $ between the "primal" productset~$\PRIMAL \times \PRIMALBIS$ and the "dual" product set $\DUAL \times\DUALBIS$. Then, we consider any bivariate function $\kernel : \PRIMAL \times\PRIMALBIS \to \barRR $ and univariate functions $\fonctionprimal : \PRIMAL\to \barRR $ and $\fonctionprimalbis : \PRIMALBIS \to \barRR $, all definedon the "primal" sets. We prove that $\fonctionprimal\np{\primal} \geq\inf\_{\primalbis \in \PRIMALBIS} \Bp{\kernel\np{\primal, \primalbis} \UppPlus\fonctionprimalbis\np{\primalbis}} $ $ \Rightarrow\SFM{\fonctionprimal}{\coupling}\np{\dual} \leq \inf\_{\dualbis \in \DUALBIS}\Bp{\SFM{\kernel}{\SumCoupling{\coupling}{\couplingbis}}\np{\dual,\dualbis}\UppPlus \SFM{\fonctionprimalbis}{-\couplingbis}\np{\dualbis}} $, where westress that the Fenchel-Moreau conjugates $\SFM{\fonctionprimal}{\coupling} $and $\SFM{\fonctionprimalbis}{-\couplingbis}$ are not necessarily taken withthe same coupling. We study the equality case, after having established theclassical Fenchel inequality but with a general coupling. % We display severalapplications. We provide a new formula for the Fenchel-Moreau conjugate of ageneralized inf-convolution. We obtain formulas with partial Fenchel-Moreauconjugates. Finally, we consider the Bellman equation in stochastic dynamicprogramming and we provide a "Bellman-like" equation for the Fenchel conjugatesof the value functions.

Fenchel-Moreau Conjugation Inequalities with Three Couplings and Application to Stochastic Bellman Equation