MULTIPLICATIVE TEMPERED FRACTIONAL INTEGRALS IN G-CALCULUS AND ASSOCIATED HERMITE–HADAMARD-TYPE INEQUALITIES

This paper introduces the first theory of tempered fractional integrals within the framework of [Formula: see text]-calculus, a multiplicative non-Newtonian system for positive-valued functions with positive arguments. We begin by formulating the multiplicative Riemann–Liouville integral in its pure multiplicative form and extend it to include an exponential tempering parameter. A new multiplicative [Formula: see text]-incomplete Gamma function is defined to characterize these operators. Furthermore, we introduce and analyze multiplicative convexity in [Formula: see text]-calculus, along with novel multiplicative formulations of the classical midpoint and trapezoidal quadrature rules. We then establish the Hermite–Hadamard inequalities for [Formula: see text]-convex functions and derive two novel multiplicative integral identities, leading to midpoint- and trapezium-type bounds. Numerical examples with graphical illustrations, applications to quadrature rules, and connections to special means validate our results. The proposed framework fills a critical gap in non-Newtonian analysis and provides new tools for modeling scale-invariant phenomena in economics, biology, and signal processing.