Abstract
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.
Highlights
The theory of convexity has become a broad area of study since it is related to the theory of inequalities
One example of these is how the convexity was applied to estimate errors when using a trapezoidal formula for numerical integration [6,7]
An interesting inequality for convex function is of Hermite–Hadamard type, which can be stated as follows: Let S be a nonempty subset in R, ψ : S → R be a convex function on S, and u1, u2 ∈ S, u1 < u2, we have u1 + u2 ψ ( u1 ) + ψ ( u2 )
Summary
The theory of convexity has become a broad area of study since it is related to the theory of inequalities. Both the positive monotone and positive convex functions belong to this class This concept has been recently extended to s-Godunova–Levin type of convexity by Dragomir [17]. Studies were conducted on s-Godunova–Levin type convexity and can be found in the literature [6] Another important class of convex function is h-convexity, which was introduced by Varošanec [18], through which several generalizations and extensions were made. We present new Hermite–Hadamard type inequalities for h-Godunova–Levin preinvexity.
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