Being a critical part of classical analysis, some of the convex functions and inequalities have drawn much attention recently because both concepts establish a strong relationship. As a familiar extension of classical one, the interval-valued analysis is frequently used to the re-search of control theory, mathematical economy and so on. Motivated by the importance of convexity and inequality, our aim is to consider new class of convex interval-valued functions is known as LR-(𝑝, 𝒽)-convex interval-valued functions through pseudo order relation(≤𝑝). This order relation is defined on interval space. By using this concept, firstly we obtain Her-mite-Hadamard (𝐻𝐻-) and Hermite- Hadamard-Fejér (𝐻𝐻-Fejér) type inequalities through pseudo order relation. Secondly, we present some new versions of discrete Jensen and Schur type inequalities via LR-(𝑝, 𝒽)-convex interval-valued functions. Moreover, we have shown that our results include a wide class of new and known inequalities for LR-𝒽-convex-IVFs and their variant forms as special cases. Under some mild restrictions, we have proved that the inclusion relation “⊆” coincident to pseudo order relation “≤𝑝” when the interval-valued function is LR-(𝑝, 𝒽)-convex or LR-(𝑝, 𝒽)-concave. Results obtained in this paper can be viewed as improvement and refinement of previously known results.