Generalization of Hermite–Hadamard, trapezoid, and midpoint Mercer type inequalities for fractional integrals in multiplicative calculus

This study generalizes Hermite–Hadamard–Mercer type inequalities using Riemann–Liouville fractional integrals within the framework of multiplicative calculus. Multiplicative fractional integral identities are established for ∗differentiable convex functions, forming the basis for deriving trapezoidal-Mercer and midpoint-Mercer type inequalities in this context. Numerical examples and graphical analysis are provided to demonstrate the applicability and effectiveness of these inequalities, along with applications to special means of real numbers. For the first time, applications to quadrature formulas for midpoint and trapezoidal rules are presented within the framework of multiplicative calculus. This work underscores the versatility of the fractional operator in addressing problems involving noninteger order differentiation, offering refinements to classical inequalities. By extending these inequalities, the research aims to uncover new mathematical insights, properties, and relationships, contributing to developing advanced mathematical tools applicable across various scientific disciplines.