Some Integral Inequalities for h-Godunova-Levin Preinvexity

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.


Introduction
Recently, the theory of convexity has become a broad area of study since it is related to the theory of inequalities. Many such inequalities are frequently reported in the literature as a result of applications of convexity in both pure and applied sciences (see [1][2][3][4]). Considering its many applications in different branches of mathematics, convexity can provide a basis for estimating error bounds in a large class of problems [5]. One example of these is how the convexity was applied to estimate errors when using a trapezoidal formula for numerical integration [6,7]. Others include studying problems in nonlinear programming and applying them to special means [8]. Among them, 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 u 1 , u 2 ∈ S, u 1 < u 2 , then we have If ψ is a concave function, the two inequalities can be held in the reverse direction. These inequalities have been extensively improved and generalized. For example, see [1,[9][10][11][12]. Definition 1. [13] A positive function ψ : S ⊆ R → R is said to be a Godunova-Levin if ψ(δu 1 + (1 − δ)u 2 ) ≤ ψ(u 1 ) δ + ψ(u 2 ) 1 − δ , ∀u 1 , u 2 ∈ S, δ ∈ (0, 1).
Several other properties related to this class of functions are given in [14][15][16]. For example, both the positive monotone and positive convex functions belong to this class.
Definition 4. [20] A function ψ on the invex set S is said to be preinvex with respect to ζ if Usually, the preinvex functions can be convexity if ζ(u 2 , u 1 ) = u 2 − u 1 holds in (2). Other properties of preinvex functions are given in [21,22].
We arrange this paper as follows. Section 2 introduces the new classes of h-Godunova-Levin, denoted by SGX( 1 h , t) and SGV( 1 h , t), together with their properties. This class of function unifies different classes of convexity: s-Godunova-Levin, P-functions, s-convexity, and Godunova-Levin. In Section 3, we prove new Hermite-Hadamard inequalities via h-Godunova-Levin preinvexity. Section 4 introduces a new definition of h-Godunova-Levin preinvexity, which can be the generalization of preinvexity. This Section also presents new Hermite-Hadamard type inequalities for h-Godunova-Levin preinvexity. Section 5 gives some applications to special means, as well as an application to numerical integration.

The h-Godunova-Levin Functions and Their Properties
This section introduces the notion of h-Godunova-Levin function together with their properties. This class of function can be denoted by SGX( 1 h , t) and SGV( 1 h , t) for h-Godunova-Levin convex and h-Godunova-Levin concave, respectively.

Definition 5.
Suppose h : (0, 1) → R. A non-negative function ψ : S → R is said to be h-Godunova-Levin, or that ψ belongs to the class SGX( 1 h , S), for all u 1 , u 2 ∈ S and δ ∈ (0, 1), we have , the definition of h-Godunova-Levin function can be clearly reduced to different types of convexity, such as Godunova-Levin function, classical convex, s-Godunova-Levin function, P-function, and s-convex function. This indicates that h-Godunova-Levin function is the generalization of these different classes.

Proposition 1.
Suppose that h 1 , h 2 are two positive functions defined on the interval S satisfying the property , δ ∈ (0, 1).

Proof. The proof is clear from the definition of the classes h-Godunova-Levin convex and
Proposition 3. Suppose that ψ and ω are two h-Godunova-Levin functions and satisfying the property given in Definition 2. Then, the product of these two functions satisfies Proof. Given that ψ and ω are h-Godunova-Levin functions, we have .

New Hermite-Hadamard Inequality for h-Godunova-Levin Convex Function
The following generalization of the Hermite-Hadamard inequalities for h-Godunova-Levin convex function can be proved in this section. Considering Thus, after integrating (6), we get the following This ends the proof of the first inequality. Now, taking v 1 = u 1 and v 2 = u 2 in (5) and integrating the result over the interval [0, 1] with respect to δ, we obtain This completes the proof of the second inequality (4).

Hermite-Hadamard Inequalities for h-Godunova-Levin Preinvex Function
The definition of h-Godunova-Levin preinvex is introduced in this section. The inequalities of Hermite-Hadamard type for functions whose first derivatives absolute values are h-Godunova-Levin preinvex are also presented here.
The following lemma can be used to prove the generalization of the Hermite-Hadamard inequalities for h-Godunova-Levin preinvexity. Lemma 1. [24] Suppose that ψ : , then we get the following inequality: dδ.

Theorem 4.
With the assumptions of Theorem 3, we get the following: Proof. We use Lemma 1 to show that Applying power-mean inequality, we get Applying the basic calculus, we have

Applications to Numerical Integration
As mentioned in the introduction, the convexity can be applied to many areas of studies. Here, we give an example of how the h-Godunova-Levin convex and preinvex functions can be used to estimate the errors accumulated when using the trapezoidal formula for numerical integration.
Let d be a division of the interval [u 1 , u 2 ], i.e., d : is the trapezoidal formula. The associated approximation error is denoted by E(ψ, d).

Applications to Special Means
We finally use Hermite-Hadamard inequalities for h-Godunova-Levin preinvex function to form the inequalities for special means. Thus, the means of two positive numbers u 1 , u 2 , and u 1 = u 2 can be considered as follows: 1.
The arithmetic mean: The generalized log-mean: The following propositions are obtained from the results in Section 4 and the above applications of special means. Proof. We derived this inequality from Corollary 3 applied to the h-Godunova-Levin preinvex function ψ : R → R.

Conclusions
Since the Hermite-Hadamard type inequalities, due to their importance, can be found in many fields of study, the present study established new generalizations of such inequalities. Thus, two classes of function, h-Godunova-Levin and h-Godunova-Levin preinvex functions, along with some of their properties were established here. The applications to special means and numerical integration were also discussed in this study.
Funding: This research received no external funding