Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures

The aim of this manuscript is to introduce the fractional integral inequalities of H-H types via multiplicative (Antagana-Baleanu) A-B fractional operators. We also provide the fractional version of the H-H type of the product and quotient of multiplicative superquadratic and multiplicative subquadratic functions via the same operators. Superquadratic functions, have stronger convexity-like behavior. They provide sharper bounds and more refined inequalities, which are valuable in optimization, information theory, and related fields. The use of multiplicative fractional operators establishes a nonlinear fractional structure, enhancing the analytical tools available for studying dynamic and nonlinear systems. The authenticity of the obtained results are verified by graphical and numerical illustrations by taking into account some examples. Additionally, the study explores applications involving special means, special functions and moments of random variables resulting in new fractional recurrence relations within the multiplicative calculus framework. These contributions not only generalize existing inequalities but also pave the way for future research in both theoretical mathematics and real-world modeling scenarios.