  Given two couplings between "primal" and "dual" sets, we prove a general
implication that relates an inequality involving "primal" sets to a reverse
inequality 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" product
set~$\PRIMAL \times \PRIMALBIS$ 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 defined
on 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 we
stress that the Fenchel-Moreau conjugates \(\SFM{\fonctionprimal}{\coupling} \)
and \(\SFM{\fonctionprimalbis}{-\couplingbis}\) are not necessarily taken with
the same coupling. We study the equality case, after having established the
classical Fenchel inequality but with a general coupling. % We display several
applications. We provide a new formula for the Fenchel-Moreau conjugate of a
generalized inf-convolution. We obtain formulas with partial Fenchel-Moreau
conjugates. Finally, we consider the Bellman equation in stochastic dynamic
programming and we provide a "Bellman-like" equation for the Fenchel conjugates
of the value functions.
