Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings

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) $\mathfrak{J}$-convex interval-valued mappings through pseudo-order relation ( ${\le }_p$) 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-Fejér (HH-Fejér) 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 $\mathfrak{J}$-convex- I-V⋅Ms and their variant forms as special cases. Under some mild restrictions, we have proved that the inclusion relation " $\subseteq$" is coincident to pseudo-order relation " ${\le }_p$" when the I-V⋅M is L-R $\mathfrak{J}$-convex or L-R $\mathfrak{J}$-concave. Results obtained in this paper can be viewed as an improvement and refinement of classical known results.