Fitzpatrick Functions: Inequalities, Examples, and Remarks on a Problem by S. Fitzpatrick

In 1988, Simon Fitzpatrick defined a new convex function FA -- nowadays called the Fitzpatrick function -- associated with a monotone operator A, and similarly a monotone operator Gf associated with a convex function f. This paper deals with two different aspects of Fitzpatrick functions. In the first half, we consider the Fitzpatrick function of the subdifferential of a proper, lower semicontinuous, and convex function. A refinement of the classical Fenchel-Young inequality is derived and conditions for equality are investigated. The results are illustrated by several examples. In the second half, we study the problem, originally posed by Fitzpatrick, of determining when A = GFA. Fitzpatrick proved that this identity is satisfied whenever A is maximal monotone; however, he also observed that it can hold even in the absence of maximal monotonicity. We propose a new condition sufficient for this identity, formulated in terms of the polarity notions introduced recently by Martínez-Legaz and Svaiter. Moreover, on the real line, this condition is also necessary and it corresponds to the connectedness of A.