On Upper Estimations of Hermite–Hadamard Inequalities

Convex functions play a key role in many branches of pure and applied mathematics. In this paper, we prove that if a convex function is not continuous, then the classical Hermite–Hadamard inequality, the Hermite–Hadamard inequality for the Riemann–Liouville fractional integral and the Hermite–Hadamard inequality for the left variable order Riemann–Liouville fractional integral can be improved with slightly sharper bounds. In particular, we give new upper bounds of Hermite–Hadamard inequalities in terms of right-hand limit ga+ and left-hand limit gb− values. Furthermore, classical Hermite–Hadamard inequalities only applied to closed bounded intervals, but our new improved inequalities can be applied to open bounded and half-open bounded intervals. As a consequence of our method, we also show some nonconvex functions that satisfy our new improvement of Hermite–Hadamard inequalities.