New Hermite–Hadamard Type Inequalities in Connection with Interval-Valued Generalized Harmonically ( h 1 , h 2 ) -Godunova–Levin Functions

: As is known, integral inequalities related to convexity have a close relationship with symmetry. In this paper, we introduce a new notion of interval-valued harmonically ( m , h 1 , h 2 ) Godunova–Levin functions, and we establish some new Hermite–Hadamard inequalities. Moreover, we show how this new notion of interval-valued convexity has a close relationship with many existing deﬁnitions in the literature. As a result, our theory generalizes many published results. Several interesting examples are provided to illustrate our results.


Introduction
Interval analysis is a subset of set-valued analysis, which is the study of sets in the context of mathematics and general topology. The Archimedean approach, which includes determining a circle's circumference, is a classic illustration of interval enclosure. This theory addresses the interval uncertainty that exists in many computational and mathematical models of deterministic real-world systems. With this approach, errors that result in incorrect conclusions are avoided by studying interval variables instead of point variables and expressing computation results as intervals. Consideration of the error estimates of the numerical solutions for finite state machines was one of the initial goals of the interval-valued analysis. Interval analysis, which Moore first described in his renowned book [1], is one of the fundamental techniques in numerical analysis. As a result, it has found applications in many fields, including differential equations for intervals [2], neural network output optimization [3], automatic error analysis [4], computer graphics [5], and many more. For results and applications, we refer interested readers to [6][7][8][9][10].
Inequalities have a significant impact on mathematics, particularly those connected to the Jensen, Ostrowski, Hermite-Hadamard, Bullen, Simpson, and Opial inequalities. Many of these inequalities have recently been extended to interval-valued functions by some well-known researchers (see, for example, [11][12][13][14]), and many have also researched the Hermite-Hadamard inequality for convex functions. The traditional H-H inequality is given as: where G is a convex function.
On the other hand, the generalized convexity of mappings is a potent tool for addressing a broad range of issues in applied analysis and nonlinear analysis, including various problems in mathematical physics. Recently, a number of generalizations of convex functions have been thoroughly researched. Mathematical analytic study on the idea of integral inequalities is interesting. The study of differential and integral equations has also been considered to be relevant for inequalities and various extended convex mappings. Electrical networks, symmetry analysis, operations research, finance, decision making, numerical analysis, and equilibrium are just a few areas where they have had a substantial impact. We investigate how the subjective properties of convexity might be encouraged by using a number of fundamental integral inequalities.
The Hermite-Hadamard inequality is related to various classes of convexity; for some examples, see [15][16][17][18][19]. Iscan [20], in 2014, presented the idea of harmonic convexity and established a few related Hermite-Hadamard type inequalities. Harmonic h-convex functions and some associated Hermite-Hadamard inequalities were first described by the authors of [21] in 2015. Numerous researchers have linked integral inequalities with interval-valued functions in recent years, producing many significant findings. The Opial-type inequalities were introduced by Costa [22], the Ostrowski-type inequalities were investigated by Chalco-Cano [23] by using the generalized Hukuhara derivative, the Minkowski-type inequalities and the Beckenbach-type inequalities were established by Roman-Flores [24]. By introducing interval-valued coordinated convex functions and creating related H − Htype inequalities, Zhao et al. [25] recently improved on this idea. It was also utilized to support the H − H-and Fejér-type inequalities for the n-polynomial convex interval-valued function [26] and preinvex function [27,28]. Interval-valued coordinated preinvex functions are a recent extension of the interval-valued preinvex function notion introduced by Lai et al. [29]. Combined with interval analysis, the H − H inequality was extended to interval h-convex functions in [30], to interval harmonic h-convex functions in [31], to interval (h 1 , h 2 )-convex functions in [32] and to interval harmonically (h 1 , h 2 )-convex functions in [33]. The definition of the h-Godunova-Levin function was utilized by the authors in [34] to take into account this inequality. Additionally, the author in [35] published a fuzzy Jensen-type integral inequality for fuzzy interval-valued functions, while the authors in [36] created a Jensen-type inequality for (h 1 , h 2 ) interval-nonconvex functions.
Our research is inspired by the strong literature and the specific articles [33,34]. The idea of interval-valued harmonically (m, h1, h2)-Godunova-Levin functions is introduced first, and new H − H-type inequalities are then constructed for the aforementioned notion. The structure of the paper is as follows: In Section 2, the introduction and the mathematical background are given. The issue and our key findings are discussed in Section 3. Section 5 contains the conclusion and future scope.

Preliminaries
We begin by introducing some of the terms, characteristics, and notations that will be utilized in the article. Let I be represented as the intervals of the collection of real numbers R.
[a] = [a, a] = {x ∈ R | a ≤ x ≤ a}, a, a ∈ R, where the real interval [a] is a closed and bounded subset of R. We call [a] positive when a > 0, and [a] is negative when a < 0. Let us denote all intervals of the set of real numbers by R I of R, all positive intervals by R + I , and all negative intervals by R − I . The inclusion" ⊆ " is defined as: Suppose λ is any real number, and [a] is an interval; then, the v[a] is given as: , the following algebraic operations hold true: Definition 1 (see [37]).
The set of all Riemann integrable interval-valued functions and real-valued functions are represented by the symbols IR [c,d] and R [c,d] , respectively.
Definition 4 (see [20]). A function G : S → R is said to be a harmonically convex function, if where ∀c, d ∈ S and u ∈ [0, 1].
Definition 5 (see [21]). The function G : S → R is said to be a harmonically h-convex function, if ∀c, d ∈ S and u ∈ [0, 1], we have where h : [0, 1] ⊆ S → R is a nonnegative function with h = 0.
If the above inequality is in reverse order, then the function G is known as a harmonically That is, It follows that This shows that G ∈ SGHX The proof can be conducted in a similar manner as in Proposition 1; hence, it is omitted for the readers.
. (2) Proof. Assume that G ∈ SGHX Then, Multiplying both sides by If we integrate the above inequality over (0, 1) with respect to "oe", we have Thus, It follows (change of variables) that As a result, we have Upon dividing both sides by 1 2 , we obtain the desired first inclusion of Theorem 1.
From our hypothesis, we have Adding the above two inclusions and integrating over (0, 1) with respect to "σ" gives It is easily seen that From the above developments, it follows Now, combining (5) and (6), we obtain the required result .
and we obtain 1 Integrating over (0, 1) with respect to "σ", we have 1 i.e., 1 and consequently, Similarly, for the interval 2cd c+d , md , we can have Adding (8) and (9), we obtain Consequently, we obtain H A minor modification of the argument in Theorem 1 will give This completes the proof.
Proof. Assume that G, V ∈ SGHX From the above hypotheses, we obtain Integrating both sides over (0, 1) with respect to "σ", we have It follows that The Theorem is proved.   [G(c)V (c) + G(d)V (d)] Integrating with respect to "σ" over (0, 1), we have , we obtain .