This study investigates a new class of totally ordered interval-valued super-quadratic convex mappings that are based on the ordering relation $\mathfrak{c}\mathfrak{r}$ and control mappings ħ. We also discuss a few captivating properties of this class, along with some special cases, to ensure its broadness and utility. We look at classical integral inequalities such as Jensen’s, reverse Jensen’s, Jensen–Mercer, and Hermite–Hadamard’s inequality along with interval-valued $\mathfrak{c}\mathfrak{r}$ -h super-quadratic mappings in the context of classical and fractional operators from a useful point of view. The outcomes are affirmed through illustrative examples, visuals, and applications to means and information theory, respectively.
