Some Novel Estimates of Integral Inequalities for a Generalized Class of Harmonical Convex Mappings by Means of Center-Radius Order Relation

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Introduction
Uncertainty problems can be distorted by using specifc numbers. It is therefore crucial to avoid such errors and obtain efective results. Furthermore, Moore [1] proposed and investigated interval analysis for the frst time in 1966. It is a discipline in which an uncertain variable is represented by an interval of real numbers. Trough this analysis, the accuracy level of problems is improved. A variety of felds have been afected by it over the past 50 years, including diferential equations with intervals [2], neural networks [3], aeroelasticity [4], and error analysis [5]. As a result, interval analysis has yielded many excellent results, and readers interested in reading more can consult reference [6].
Te concept of convexity is becoming increasingly important to both the pure and applied sciences. Research on the concept of convexity with integral problems is an exciting area. Integral inequalities are useful for evaluating convexity and nonconvexity qualitatively and quantitatively. Tis area of research has grown in popularity due to its diverse applications in diferent felds. Convexity is an integral part of optimization concepts and is widely used in operation research, economics, control theory, decision-making, and management. Tere is a great deal of experience among mathematicians who deal with inequalities, such as those related to Ostrowski, Opial, Simpson, Jensen, and Hermite− Hadamard. In an attempt to promote convexity subjectively, we apply several fundamental integral inequalities. Tis has resulted in many inequalities as an application of convex functions and generalized convex functions, see reference [7]. Later, various partial order relations, as well as diferent integral operators, were used to establish a strong interrelationship between inequalities and interval-valued functions (IVFS). Khan et al. [8] established Hermite− Hadamard type inequalities for left right interval-valued functions. Nwaeze et al. [9] developed a fractional version of these inequalities for polynomial convex interval-valued functions. Several practical applications have been developed based on these concepts see reference [10]. For IVFS, initially, Breckner introduces the concept of continuity, see reference [11]. Chalco-Cano et al. [12] established the Ostrowski-type inequality, Costa et al. [13], Flores-Franulic and Román-Flores [14], and Costa et al. [13] established the Opial-type inequality for IVFS. A famous double inequality is defned as follows: An approximation of the mean value of a continuous function is provided by the function. Despite its simplicity, it is well known because of its defnition of convex mappings, the frst geometrical interpretation in elementary mathematics. In 2007, Varoşanec [15] introduced the idea of h-convexity. Inspired by this idea, Zhao et al. [16], introduced the notion of h-convexIVFS, and utilizing these notions, Jensen and H.H inequality were established. Tere are a variety of interval-valued generalizations of these inequalities. Zhang et al. [17] developed Hermite− Hadamard and Jensen-type inequalities for the generalized class of Godunova-Levin functions. Afzal et al. [18] introduced the notion of Harmonical Godunova− Levin intervalvalued functions and developed these inequalities see reference [19]. Initially, Awan et al. introduced the notion of (h 1 , h 2 )-convex functions and developed the following results [20]. Later, diferent authors used the notion of (h 1 , h 2 )-convexity and developed the following inequalities using related classes of convexity see references [21][22][23]. Te results developed using partial order relations, including inclusion relations, pseudoorder relations, and fuzzy order relations are not as accurate as the results developed using the center-radius method. Terefore, center-radius order is an ideal tool for studying inequalities, and it was developed by the following author [24]. Based on harmonical CR-hconvex functions, this result can be proved by Liu et al. [25].
Te set of all harmonically CR-h-convex functions over In addition, Jensen-type inequality was also established using the notion of harmonical h-convexity via center-radius order relation.
Theorem 2 (See [25]). Let c i ∈ R + , j i ∈ [q, r]. If h is nonnegative super multiplicative function and Ψ ∈ SHX(CR − h, [q, r], R + I ) then this holds: Tere is novelty and signifcance in this study because for the frst time, harmonical (h 1 , h 2 )-convexity is connected with center-radius order relations. Furthermore, this class is more generalized since diferent choices of h result in diferent classes of harmonic convex functions. Tere are a variety of partial order relations, but CR-order is distinct from them. Te center and radius concepts can be calculated by using the endpoints of intervals such as: q C � q +q/2 and q R � q − q/2, respectively, where q � [q, q].
We get our research ideas from the extensive literature and specifc articles, see references [21,25]. Using the notions of harmonical convexity and center-radius order, we introduce a novel class of convexity called harmonical CR-(h 1 , h 2 )-convex functions. By utilizing this new idea, we developed H.H and Jensen-type inequalities. Furthermore, the study provides relevant examples to back up its fndings.

Preliminaries
Tis section summarizes some fundamental concepts, results, and defnitions. Several terms were mentioned but not explained, see references [16,25]. As you proceed through the paper, it will prove very helpful to have a basic understanding of interval analysis arithmetic 2 Journal of Mathematics where ] ∈ R. Suppose R I and R + I be the collection of all closed and positive intervals of R, respectively. We will now talk about certain interval arithmetic algebraic properties.
Let q � [q, q] ∈ R I , then q c � q + q /2 and q r � q − q /2 are the center and radius of interval q, respectively. Te centerradius form of interval q can be represented as follows: Defnition 1. Te CR-order relation for q � [q, q] � 〈q c , q r 〉 and r � [r, r] � 〈r c , r r 〉 ∈ R I can be represented as follows: NOTE: For any arbitrary two intervals q, r ∈ R I , this holds either q⪯ CR r or r⪯ CR q. Riemann integral operators for IVFS are represented as follows: Defnition 2 (See [25]). Let P: [q, r] be an interval-valued function (IVF) such that P � [P, P]. Ten, P is Riemann integrable (IR) on [q, r] if P and P are IR on [q, r], that is as follows: Te bundle of all Shi et al. [25] demonstrated that the integral retains order on the basis of CR-order relations.
Te following example will help to prove the abovementioned theorem.
Now, again using the Defnition 1, we have

Now let us introduce the notion of harmonically CRconvexity.
Defnition 5 (See [25] If in (14) Our next step will be to defne a novel defnition for harmonically CR-(h 1 , h 2 )-convex functions.

Some Variants of Hermite− Hadamard Inequalities for Harmonical-Convex Mappings Using Total Order Relations
By integrating of the above inequality over (0,1), we have Journal of Mathematics By Defnition 6, we have By integrating of the abovementioned inequality over (0, 1), we have

Conclusions
In this article, we developed the notion of harmonically center-radius order (h 1 , h 2 )-convex mappings. By using these notions, we developed H.H and Jensen-type inequalities. In comparison with other order relations, this order produces much better results. Moreover, we generalize some recently developed results, see reference [25]. Furthermore, the study provides relevant examples to back up its fndings. Tese ideas can be used to take convex optimization in a new direction. Interval integral operators and integral inequalities studied in our study will expand the potential applications of integral inequalities in practice due to the widespread use of integral operators in engineering and other applied sciences, including diferent kinds of mathematical modeling. Various integral operators are appropriate for diferent practical problems. It is anticipated that this concept will be benefcial to other researchers working in a range of scientifc disciplines.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.
Journal of Mathematics 13