Generalized Hermite--Jensen--Mercer-Type Inequalities in Fractal Sense

Abstract

The most notable inequality pertaining convex functions is Jensen’s inequality which has tremendousapplicationsinseveralfields. MercerintroducedanimportantvariantofJensen’s inequality called as Jensen-Mercer’s inequality. Fractal sets are useful tools for describing theaccuracyofinequalitiesinconvexfunctions. WeestablishageneralizedJensen–Mercer inequality and generalized Hermite-Hadamard–Mercer inequalities for a generalized convexfunctiononareallinearfractalsetRς (0<ς ≤1). Further,wealsodemonstratesome generalized Jensen–Mercer type inequalities by employing local fractional calculus. We establish two new lemmas involving local fractional integrals. By using these lemmas, we obtain several results related to generalized Hermite–Hadamard–Mercer type integral inequalities for local differentiable generalized convex functions on real linear fractal space. Lastly, some applications related to Jensen–Mercer inequality, ς-type special means, and probability density functions are given. Thepresentapproachisefficient,reliable,andmay motivate further research in this area.