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Interval analysis distinguishes between inclusion relation and order relation. Under the inclusion relation, convexity and nonconvexity contribute to different kinds of inequalities. The construction and refinement of classical inequalities have received a great deal of attention for many classes of convex as well as nonconvex functions. Convex theory, however, is commonly known to rely on Godunova-Levin functions because their properties enable us to determine inequality terms more precisely than those obtained from convex functions. The purpose of this study was to introduce a ($ \subseteq $) relation to established Jensen-type and Hermite-Hadamard inequalities using $ (h_1, h_2) $-Godunova-Levin interval-valued functions. To strengthen the validity of our results, we provide several examples and obtain some new and previously unknown results.