An innovative perspective on fractional inequalities through fractional operators and extended convexity

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.