Some Aspects of the Representation of c-Monotone Operators by C-Convex Functions

Several takes on generalizing the theory of representation of monotone operators by convex functions to the full generality of c-convexity have been proposed in recent years. In particular, given a monotone operator, a new family of convex antiderivatives is now associated with it, both in classical convex analysis as well as in the generality of c-convexity. In the present paper we take the generalization of the theory to c-convexity a few steps farther. In particular, we study the C-convex separable representation in detail, construct the sequence of Fitzpatrick functions of higher orders and present its basic properties in the generality of c-convexity, and, finally, we present a new example that demonstrates why the associated family of antiderivatives is a more natural environment for the Fitzpatrick function in an even more dramatic manner than in the classical case: the Fitzpatrick function turns out to be the maximal(!) member of the Fitzpatrick family, although it is still a minimal convex antiderivative.