This paper presents the development of reverse Minkowski and reverse Hölder's fractional integral inclusions. We propose a generic class of ${\eta }_{\hslash }$ interval-valued $(\mathrm{I}.\mathrm{V})$ convex functions, which unifies various existing classes. Additionally, we obtain a discrete Jensen-type inclusion within this convexity setup. By leveraging this advanced convexity structure together with tempered fractional integral operators, we derive new Hermite–Hadamard ( $\mathrm{H}$- $\mathrm{H}$)-type, Fejér- $\mathrm{H}$- $\mathrm{H}$-type, and other fractional inclusions. Moreover, we explore the broader significance of our results, supporting them with graphical visualizations. The applications of our results are demonstrated through average value computations.

The theory of integral inequalities is significantly advanced by the relationship between fractional calculus and convexity. The current study used extended convex functions and Katugampola fractional operators to derive Hermite-Hadamard, Fejér-Hermite-Hadamard-type inequalities as well as a few other fractional integral inequalities. We used two extended-type convex functions: log-convex and exponentially trigonometric convex functions. Our conclusions were supported by tabular data and graphical representations, which offer numerical and visual validation of the explored results. This study improves mathematical analysis and expands the scope of these inequalities by emphasizing their applicability across different forms of convexity. The knowledge acquired is highly valuable for theoretical investigation as well as real-world applications in a variety of scientific fields.