Advancements in conticrete Hermite–Hadamard–Mercer type fractional inequalities via separable sequences

Abstract The Hermite–Hadamard inequality remains a central focus in mathematical research, with its profound significance continually motivating mathematicians to explore new areas for its enhancement and generalizations. The main objective of this paper is to introduce a novel approach to establish conticrete forms of the Hermite–Hadamard–Mercer type inequalities in fractional framework. A broader notion than synchronous and monotonic sequences known as separable sequences is employed in connection with convexity theory and Riemann–Liouville fractional operators to establish these inequalities. These inequalities are constructed by employing three ξ -sequences and dual bases in R ξ ${\mathbb{R}}^{\xi }$ . These results are further expanded by a number of corollaries utilizing various sequences and bases. The remarks presented at the end of the given corollaries exhibit applications of the primary result for synchronous sequences, monotonic sequences, nondecreasing sequences in P -mean, star-shaped sequences and convex sequences. Also, new and old inequalities are established as special cases of the major findings. Finally, the paper demonstrates applications of the main outcomes to special means.