
Journal of Convex Analysis 13 (2006), No. 3, 499523 Copyright Heldermann Verlag 2006 Fitzpatrick Functions: Inequalities, Examples, and Remarks on a Problem by S. Fitzpatrick Heinz H. Bauschke Dept. of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 hbauschk@uoguelph.ca D. Alexander McLaren Dept. of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 amclaren@uoguelph.ca Hristo S. Sendov Dept. of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 hssendov@uoguelph.ca In 1988, Simon Fitzpatrick defined a new convex function F_{A}  nowadays called the Fitzpatrick function  associated with a monotone operator A, and similarly a monotone operator G_{f} 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 FenchelYoung 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 = G_{FA}. 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 MartínezLegaz and Svaiter. Moreover, on the real line, this condition is also necessary and it corresponds to the connectedness of A. Keywords: Convex function, Fenchel conjugate, FenchelYoung inequality, Fitzpatrick function, monotone operator, monotone set. MSC: 26B25, 47H05; 47H04, 52A41, 90C25 [ Fulltextpdf (647 KB)] for subscribers only. 