An expanded analysis of multiplicative integral inequalities in  <i>G</i> -calculus

Abstract In this paper, we develop a framework for multiplicative quadrature rules and integral inequalities within the context of G -calculus. After reviewing the fundamental concepts of multiplicative derivatives and integrals, we introduce multiplicative analogues of classical quadrature formulas, specifically, the midpoint, trapezoidal, Simpson, and Bullen rules. We identify GG -convexity as the natural counterpart of classical convexity in this setting and introduce the notion of functions of bounded multiplicative variation. Based on these foundations, we establish the multiplicative Hermite–Hadamard inequality for GG - <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="fraktur">h</m:mi> </m:math> $\mathfrak{h}$ -convex functions. A new parametrized multiplicative integral identity is derived and employed to prove a family of parametrized inequalities for *-differentiable GG - <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="fraktur">h</m:mi> </m:math> $\mathfrak{h}$ -convex functions, functions with bounded *-derivatives, and functions of bounded *-variation. By choosing appropriate parameter values, we recover various known and new multiplicative inequalities of midpoint, trapezium, Simpson, and Bullen type. The validity of our results is verified through two numerical examples with graphical illustrations. Applications to special means are also presented, and the paper concludes with a summary of the main findings and potential directions for future work.