This work utilized the concepts of fuzzy interval analysis and convexity to explore some novel refinements of classical counterparts. The main goal was to look into a type of strong convexity that connected the ideas of pseudo-ordering, ${\delta }^1$-level mappings, and the control function ${\hslash }_{\circ }$. This type of mapping is called a fuzzy number-valued ${\hslash }_{\circ }$-super-quadratic mapping. An interesting fact is that all the function classes extracted from this class were new and novel and quite useful in the optimization and approximation theory. We assessed this class of functions pertaining to essential properties, examples, and various integral inequalities such as Jensen's, reverse Jensen's, Jensen-Mercer, Hermite-Hadamard and Fejer's like inequalities in the classical, and fractional framework. Furthermore, we delivered the accuracy of our findings through graphical and tabular approaches, particularly a novel application for means.