Abstract
The notions of convex mappings and inequalities, which form a strong link and are key parts of classical analysis, have gotten a lot of attention recently. As a familiar extension of the classical one, interval- valued analysis is frequently used in the research of control theory, mathematical economy and so on. Motivated by the importance of convexity and inequality, our aim is to consider a new class of convex interval-valued mappings (I-V.Ms) known as left and right (L-R) Z-convex interval-valued mappings through pseudo-order relation (<=(p)) or partial order relation, because in interval space, both concepts coincide, so this order relation is defined in interval space. By using this concept, first we obtain Hermite-Hadamard (HH-) and Hermite-Hadamard-Fejer (HH-Fejer) type inequalities through pseudo-order relations via the Riemann-Liouville fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for L-R Z-convex- I-V.Ms and their variant forms as special cases. Under some mild restrictions, we have proved that the inclusion relation "subset of" is coincident to pseudo-order relation "<=(p)" when the I-V.M is L-R Z-convex or L-R Z-concave. Results obtained in this paper can be viewed as an improvement and refinement of classical known results.