Generalized pseudo-integral Jensen's inequality for ((⊕1,⊗1),(⊕2,⊗2))-pseudo-convex functions

Abstract

It is remarked that the generalization of Jensen's inequality for pseudo-integrals (Pap and Štrboja [14]) is not a complete generalization of the classical Jensen's inequality, and a generalized Jensen's inequality for pseudo-integral with respect to (⊕,⊗)-pseudo-convex function is given in [28]. The present article is a continuation of the previous work. A new notion of (⊕1,⊗1),(⊕2,⊗2)-pseudo-convex function is introduced, which generalize the notion of (⊕,⊗)-pseudo-convex function and many other previously generalizations. Motivated by the work of Kaluszka et al. [6], related to Jensen's inequality with respect to different generalized fuzzy integrals, a new generalized Jensen's inequality between a pseudo-integral and general fuzzy integral, as well as between two different pseudo-integrals, with respect to ((⊕1,⊗1),(⊕2,⊗2))-pseudo-convex functions are proved. These results cover all previously obtained Jensen's inequalities for pseudo-integrals (Zhang and Pap [28]) as well as the classical Jensen's inequality.