Jensen's inequalities for set-valued and fuzzy set-valued functions

Abstract

Being an important part of classical analysis, Jensen's inequality has drawn much attention recently. Due to its generality, the inequality based on non-additive integrals appears in many forms, such as Sugeno integrals, Choquet integrals and pseudo-integrals. As a well-known generalization of classical one, the set-valued analysis is frequently applied to the research of mathematical economy, control theory and so on. Thus, it is of great necessity to generalize the set-valued case. Motivated by the pioneering work of Costa's Jensen's fuzzy-interval-valued inequality and Štrboja et al.'s Jensen's set-valued inequality based on Aumann integrals and pseudo-integrals respectively, this paper focuses particularly on proving certain kinds of Jensen's set-valued inequalities and fuzzy set-valued inequalities. These inequalities consist of two families: the related convex (or concave) function is a set-valued or fuzzy set-valued function and the integrand is a real-valued function; the related convex (or concave) function is a real-valued function and the integrand is a set-valued or fuzzy set-valued function. Particularly, Jensen's interval-valued and fuzzy-interval-valued inequalities, including Costa's, are obtained as corollaries.