Exploring fractal–fractional integral inequalities: An extensive parametric study

Abstract

In this paper, we investigate fractal–fractional integral inequalities for generalized $(s,P)$-convex functions, a topic of growing interest in the field of fractional calculus. We begin by establishing a fractal–fractional Hermite–Hadamard inequality, providing a novel perspective on fractal $(s,P)$-convexity. Subsequently, we introduce a parameterized identity involving fractal–fractional integrals, which serves as a cornerstone for deriving midpoint-, trapezium-, Bullen-, Milne-, and Simpson-type inequalities. The results are developed for mappings whose fractal derivatives display generalized $(s,P)$-convexity. Additionally, we present a numerical example with graphical representations to validate the theoretical findings. By leveraging improved versions of the Hölder and power mean inequalities, we further extend the applicability of our results. The study concludes by highlighting potential applications and proposing directions for future research, emphasizing the significance of these contributions to the broader field of mathematical analysis and optimization.