Generalizing comonotonicity: New insights and dual perspectives

Abstract

We review, exemplify, and compare recently introduced classes of functions that generalize comonotone functions from two perspectives: algebraically through ${\star }_{\inf }$-associated functions and measure-theoretically via m-positively dependent functions, along with their dual versions – ${\star }_{\sup }$-associated functions and m-subadditive functions, respectively. While the concept of m-subadditive functions is well-documented in the literature, the concept of ${\star }_{\sup }$-associatedness appears to be novel. Furthermore, we present several applications of these function classes, particularly concerning integral inequalities for the generalized Sugeno integral. Additionally, we mention other extensions of comonotonicity, including weak comonotonicity and multivariate comonotonicity.