New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with ψk-Raina’s Fractional Integrals for Differentiable Convex Functions

Starting from ψk-Raina’s fractional integrals (ψk-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve ψk-RFIs. The results of the research are refinements of the Hermite–Hadamard (H-H) and H-H-M-type inequalities. For several types of fractional integrals—Riemann–Liouville (R-L), k-Riemann–Liouville (k-R-L), ψ-Riemann–Liouville (ψ-R-L), ψk-Riemann–Liouville (ψk-R-L), Raina’s, k-Raina’s, and ψ-Raina’s fractional integrals (ψ-RFIs)—new inequalities of H-H and H-H-M-type are established, respectively. This article presents special cases of the main results and provides numerous examples with graphical illustrations to confirm the validity of the results. This study shows the efficiency of the findings with a couple of applications, taking into account the modified Bessel function and the q-digamma function.