Visualizing Fractional Integral Inequalities Using Euler’s Beta Function and Extended Convexity

In this research article, we present various extensions and refinements of Hermite–Hadamard and related fractional integral inequalities by utilizing the unique characteristics of Euler’s beta and extended convex functions. In some of these results, Euler’s beta function is used as a weight function, while in the others, Euler’s incomplete beta function is employed as a weight. Corresponding to the main results, corollaries and graphical illustrations are provided to support and validate them. Our findings deepen the understanding of the interplay between fractional calculus, extended convex functions, and special functions, while building upon and enhancing recent developments. These inequalities hold potential applications across diverse fields including physics, engineering, and mathematics.