Refinement of Integral Inequalities and Their Applications to Special Functions via Fractional Calculus

In this manuscript, we take into account the notion of ${\mathfrak{h}}$-superquadraticity and employ it to derive novel Fej\'{e}r and (Hermite-Hadamard) $\mathbb{HH}$'s type inequalities using Atangana-Baleanu fractional integral operators. Since superquadraticity is a refinement of convexity, the resulting inequalities are sharper and more general than those based solely on convexity. A key advantage of $\mathfrak{h}$-superquadraticity is its flexibility; by choosing different forms for the function $\mathfrak{h}$, such as $\mathfrak{h}(\varkappa)=\varkappa$, $\mathfrak{h}(\varkappa)=\varkappa^\mathfrak{s}$, $\mathfrak{h}(\varkappa)=\frac{1}{\varkappa}$, and $\mathfrak{h}(\varkappa)=1$, we derive Fej\'{e}r and $\mathbb{HH}$-type inequalities for superquadratic functions, s-superquadratic functions, superquadratic functions of the Godunova-Levin type, and P-superquadratic functions, respectively. To validate our theoretical results, we present numerical computations, graphical representations, and several illustrative examples. Additionally, we explore applications of the obtained inequalities in the context of special functions, including Bessel and Mittag-Leffler functions by deriving new recurrence relations of fractional order that cannot be obtained through conventional analytical methods.