ON SOME NEW HERMITE–HADAMARD–MERCER-TYPE INEQUALITIES THROUGH SEPARABLE SEQUENCES AND CAPUTO FRACTIONAL OPERATORS

The Hermite–Hadamard inequality obtained a prominent place in the field of mathematical inequalities since its discovery. This inequality has been inspiring mathematicians to continuously generalize, improve and refine it. This paper explores a novel approach of establishing conticrete forms of the Hermite–Hadamard–Mercer-type inequalities within the framework of Caputo fractional derivative operators through the application of separable sequences. The main results of this paper are attributed to the derivation of conticrete Hermite–Hadamard–Mercer-type inequalities by considering three [Formula: see text]-tuples and using the property of convexity within the framework of fractional calculus. By employing various vectors, bases, and their dual bases, we derive a series of subsequent corollaries from the main inequalities. These derivations yield both new and previously established inequalities as special cases of the primary results. The remarks at the end of these corollaries extend these results to diverse sequence types, including nondecreasing sequences in P-mean, star-shaped, monotonic, synchronous and convex sequences, illustrating the broad utility of the proposed methodology.