New insights on fractal–fractional integral inequalities: Hermite–Hadamard and Milne estimates

Abstract

This paper investigates fractal–fractional integral inequalities for generalized $s$-convex functions. We begin by establishing a fractal–fractional Hermite–Hadamard inequality for such functions. In addition, a novel identity is introduced, which serves as the basis for deriving some fractal–fractional Milne-type inequalities for functions whose first-order local fractional derivatives exhibit generalized $s$-convexity. Subsequently, we provide additional results using the improved generalized Hölder and power mean inequalities, followed by a numerical example with graphical representations that confirm the accuracy of the obtained results. The study concludes with several applications to demonstrate the practicality and relevance of the proposed inequalities in various settings.