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Lower Bounds on the Haraux Function

The Haraux function is an important tool in monotone operator theory and its applications. One of its salient properties for a maximally monotone operator is to be valued in $[0,+\infty]$ and to vanish only on the graph of the operator. Sharper lower bounds for this function have been proposed in specific cases. We derive lower bounds in the general context of set-valued operators in reflexive real Banach spaces. These bounds are new, even for maximally monotone operators acting on Euclidean spaces, a scenario in which we show that they can be better than existing ones. As a by-product, we obtain lower bounds on the Fenchel--Young function in variational analysis. Several examples are given and applications to composite monotone inclusions are discussed.