On a New Generalized Convexity and Its Related Midpoint-, Trapezoid-, and Simpson-Type Inequalities via Multiplicative Calculus

Abstract

In this paper, we introduce and investigate a refined generalization of convexity, termed the multiplicatively strongly s-convex function, and establish several of its fundamental algebraic properties. Within the framework of multiplicative calculus, this generalized convexity structure is employed to derive new midpoint-, trapezoid-, and Simpson-type inequalities, thereby providing unified refinements of several classical integral inequalities. Particular choices of the underlying parameters yield a variety of meaningful special cases and corresponding novel inequalities. Furthermore, we demonstrate the applicability of the established results through inequalities involving several important special means, including the arithmetic, geometric, logarithmic, and m-logarithmic means. To complement the theoretical developments, selected consequences are illustrated numerically and graphically, providing comchosenational evidence for the validity and effectiveness of the proposed inequalities. The results contribute to the development of multiplicative convexity theory and offer a flexible framework for deriving further inequalities within multiplicative calculus.