Lakhdari, AbdelghaniSaleh, WedadBudak, HuseyinMeftah, BadreddineJarad, Fahd2026-02-052026-02-0520260218-348X1793-6543https://doi.org/10.1142/S0218348X26500246https://hdl.handle.net/20.500.12416/15847This paper introduces the first theory of tempered fractional integrals within the framework of G-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 lambda-incomplete Gamma function is defined to characterize these operators. Furthermore, we introduce and analyze multiplicative convexity in G-calculus, along with novel multiplicative formulations of the classical midpoint and trapezoidal quadrature rules. We then establish the Hermite-Hadamard inequalities for GG-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.eninfo:eu-repo/semantics/closedAccessG-CalculusFractional CalculusMultiplicative Riemann-Liouville IntegralsMultiplicative Tempered IntegralsGG-ConvexityGA-ConvexityArticle10.1142/S0218348X26500246