A Note on Carlier Inequality

TL;DR

This paper extends Carlier's quantitative Fitzpatrick inequality from Hilbert spaces to reflexive Banach spaces and improves the strong Fitzpatrick inequality in the Hilbert space setting.

Contribution

It generalizes Carlier's inequality to a broader class of spaces and enhances existing inequalities in Hilbert spaces.

Findings

Extended Carlier's inequality to reflexive Banach spaces.

Obtained an improved strong Fitzpatrick inequality in Hilbert spaces.

Bridged the gap between Hilbert and Banach space inequalities.

Abstract

Recently, Carlier established in [3] a quantitave version of the Fitzpatrick inequality in a Hilbert space. We extend this result by Carlier to the framework of reflexive Banach spaces. In the Hilbert space setting, we obtain an improved version of the strong Fitzpatrick inequality due to Voisei and Z\u{a}linescu.