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

Abstract

The most notable inequality pertaining convex functions is Jensen’s inequality which has tremendous applications in several fields. Mercer introduced an important variant of Jensen’s inequality called as Jensen-Mercer’s inequality. Fractal sets are useful tools for describing the accuracy of inequalities in convex functions. We establish a generalized Jensen–Mercer inequality and generalized Hermite-Hadamard–Mercer inequalities for a generalized con- vex function on a real linear fractal set Rς ( 0 < ς ≤ 1). Further, we also demonstrate some 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 in- equalities 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. The present approach is efficient, reliable, and may motivate further research in this area.