Some novel inequalities for LR‐h‐convex interval‐valued functions by means of pseudo‐order relation

Abstract

In both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and integral inequality is strong. By the importance of these concepts, we have introduced the new class of generalized convex function is known as LR‐ ‐convex interval‐valued function (LR‐ ‐convex‐IVF) by means of pseudo‐order relation (≤p). This order relation is defined on interval space. Under the new concept, first, both discrete and continuous new versions of Jensen‐type inequalities are presented by means of pseudo‐order relation. Second, several new Hermite–Hadamard (HH)‐ and Hermite–Hadamard–Fejér (HH‐Fejér)‐type inequalities are also derived for LR‐ ‐convex‐IVFs. Moreover, we have shown that our results include a wide class of new and known inequalities for LR‐ ‐convex‐IVFs and their variant forms as special cases. Useful examples that verify the applicability of the theory developed in this study are presented. It the end, we have proved that the set inclusion “⊆” coincident to pseudo‐order relation “≤p.” The concepts and techniques of this paper may be the starting point for further research in this area and used as a tool to investigate the research of probability and optimization, among others.