Some Hermite-Hadamard-Mercer type inequalities via Atangana-Baleanu-conformable fractional operator

This paper investigated new Hermite-Hadamard-Mercer type inequalities associated with an Atangana-Baleanu-conformable fractional integral operator. By combining the structural features of Atangana-Baleanu fractional integrals and conformable kernels, we derived a fractional framework that contained both local and nonlocal effects. A fundamental identity was first established, transforming a symmetric combination of endpoint values and fractional integral terms into a weighted integral involving the first derivative. Based on this identity and Jensen-Mercer's inequality, several new bounds were obtained under convexity assumptions on $|f^{\mathrm{\prime }}|$ and $|f^{\mathrm{\prime }}{|}^q$, where $q>1$. The results extended known Hermite-Hadamard-Mercer inequalities and reduced to classical or fractional special cases under suitable parameter choices. The proposed approach provided a flexible tool for fractional integral inequalities and related estimates in convex analysis.