We introduce the concept of interval $(h_1,h_2)$ -convex functions. Under the new concept, we establish some new interval Hermite-Hadamard type inequalities, which generalize those in the literature. Also, we give some interesting examples.

In this paper, the definition of the harmonically $cr$-h-convex function is given, and its important properties are discussed. Jensen type inequality, Hermite–Hadamard type inequalities and Fejér type inequalities for harmonically $cr$-h-convex functions are also established. In addition, some numerical examples are given to verify the accuracy of the results.

This paper is devoted to derive optimality conditions and duality theorems for interval-valued optimization problems based on gH-symmetrically derivative. Further, the concepts of symmetric pseudo-convexity and symmetric quasi-convexity for interval-valued functions are proposed to extend above optimization conditions. Examples are also presented to illustrate corresponding results.

This paper is devoted to establishing some Hermite–Hadamard-type inequalities for interval-valued functions using the coordinated h-convexity, which is more general than classical convex functions. We also discuss the relationship between coordinated h-convexity and h-convexity. Furthermore, we introduce the concepts of minimum expansion and maximum contraction of interval sequences. Based on these two new concepts, we establish some new Hermite–Hadamard-type inequalities, which generalize some known results in the literature. Additionally, some examples are given to illustrate our results.

In this paper, we present the gH-symmetrical derivative of interval-valued functions and its properties. In application, we apply this new derivative to investigate the Karush–Kuhn–Tucker (KKT) conditions of interval-valued optimization problems. Meanwhile, some examples are worked out to illuminate the obtained results.