HERMITE–HADAMARD-TYPE FRACTIONAL INTEGRAL INEQUALITIES FOR MULTIPLICATIVELY (h,m)-PREINVEX INTERVAL-VALUED FUNCTIONS

Based on the pseudo-order relation, we develop a novel family of functions called multiplicatively [Formula: see text]-preinvex interval-valued functions (MP[Formula: see text]-IVFs) and propose some interesting properties. Afterwards, we introduce the multiplicative [Formula: see text]-Riemann–Liouville fractional integral operator on interval space. Using this operator, we derive the inequalities of Hermite–Hadamard-type for MP[Formula: see text]-IVFs. In addition, some new Hermite–Hadamard-type multiplicative integral inequalities are established for multiplicatively [Formula: see text]-convex functions, multiplicatively [Formula: see text]-convex IVFs and multiplicatively [Formula: see text]-preinvex interval-valued functions. Most importantly, these inequalities not only generalize a lot of previous work, but are especially beneficial to the further study of functions with special properties on interval space. Finally, our results are confirmed by numerical estimates and graphical representations of some appropriate examples.