New Hermite-Hadamard Type Integral Inequalities for Generalized Preinvex Functions via Yang Local Fractional Calculus

Abstract

In this manuscript, we establish new local fractional integral inequalities of Hermite-Hadamard type for generalized preinvex functions in the fractal field [Formula: see text], where [Formula: see text] denotes the fractal dimension. The main results are obtained via a new local fractional integral identity involving the invex trajectories [Formula: see text] and [Formula: see text]. By combining this identity with generalized preinvexity, the generalized Hölder inequality, and a generalized power-mean integral inequality, we obtain several estimates for functions whose local fractional derivatives, as well as powers of their absolute values, are generalized preinvex. A distinctive feature of our approach is that the proofs do not require the classical Condition [Formula: see text] typically imposed on the bifunction [Formula: see text] in related literature. We discuss several particular cases, showing that known inequalities for generalized convex functions are recovered when [Formula: see text]. Finally, as applications, we obtain estimates involving generalized probability density functions, cumulative distribution functions, and expected values within the framework of Yang’s local fractional calculus.