In this study, we introduced new integral inequalities of Hermite-Hadamard type via s-convexity and studied their properties. The absolute form of the first and second derivatives for the new inequalities is considered to be s-convex. As an application, the inequalities were applied to the special means of real numbers. We give the error estimates for the midpoint formula.

In this study, we define new classes of convexity called h-Godunova-Levin and h-Godunova-Levin preinvexity, through which some new inequalities of Hermite-Hadamard type are established. These new classes are the generalization of several known convexities including the s-convex, P-function, and Godunova-Levin. Further, the properties of the h-Godunova-Levin function are also discussed. Meanwhile, the applications of h-Godunova-Levin Preinvex function are given.

In this study, we establish a new integral inequalities of Hermite-Hadamard type for s-convexity via Katugampola fractional integral. This generalizes the Hadamard fractional integrals and Riemann-Liouville into a single form. We show that the new integral inequalities of Hermite-Hadamard type can be obtained via the Riemann-Liouville fractional integral. Finally, we give some applications to special means.

In this article, we establish new Hermite–Hadamard-type inequalities via Riemann–Liouville integrals of a function ψ taking its value in a fractal subset of R and possessing an appropriate generalized s-convexity property. It is shown that these fractal inequalities give rise to a generalized s-convexity property of ψ . We also prove certain inequalities involving Riemann–Liouville integrals of a function ψ provided that the absolute value of the first or second order derivative of ψ possesses an appropriate fractal s-convexity property.

This article deals with analysing the positivity, monotonicity and convexity of the discrete nabla fractional operators with exponential kernels from the sense of Riemann and Caputo operators. These operators are called discrete nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional operators. Further, some of our results concern sequential nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional differences, such as ∇aCFRμ∇bCFCυh(x), for various values of start points a and b, and for orders υ and μ in different ranges. Three illustrative examples of the main lemmas in the case of Riemann–Liouville are given at the end of the article.