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.

The main purpose of this paper is to study certain inequalities forcr-log-h-convex functions with an interval value. To this end, we first give a definition ofcr-log-h-convexity of interval-valued functions under thecr-order and study some properties of such functions. On this basis, we establish the Jensen-, Hermite–Hadamard-, and Fejér-type inequalities forcr-log-h-convex functions, and discuss some special cases. In addition, we give some numerical examples to illustrate the accuracy of the results obtained.

In this paper, firstly we prove the relationship between interval h-convex functions and interval harmonically h-convex functions. Secondly, several new Hermite–Hadamard type inequalities for interval h-convex functions via interval Riemann–Liouville type fractional integrals are established. Finally, we obtain some new fractional Hadamard–Hermite type inequalities for interval harmonically h-convex functions by using the above relationship. Also we discuss the importance of our results and some special cases. Our results extend and improve some previously known results.