Analytical Properties and Hermite–Hadamard Type Inequalities Derived from Multiplicative Generalized Proportional σ-Riemann–Liouville Fractional Integrals

This paper investigates the analytical properties of multiplicative generalized proportional σ-Riemann–Liouville fractional integrals and the corresponding Hermite–Hadamard-type inequalities. Central to our study are two key notions: multiplicative σ-convex functions and multiplicative generalized proportional σ-Riemann–Liouville fractional integrals, both of which serve as the foundational framework for our analysis. We first introduce and examine several fundamental properties of the newly defined fractional integral operator, including continuity, commutativity, semigroup behavior, and boundedness. Building on these results, we derive a novel identity involving this operator, which forms the basis for establishing new Hermite–Hadamard-type inequalities within the multiplicative setting. To validate the theoretical results, we provide multiple illustrative examples and perform graphical visualizations. These examples not only demonstrate the correctness of the derived inequalities but also highlight the practical relevance and potential applications of the proposed framework.