Abstract
In interval analysis, integral inequalities are determined based on different types of order relations, including pseudo, fuzzy, inclusion, and various other partial order relations. By developing a link between center-radius (CR) order relations, it seeks to develop a theory of inequalities with novel estimates. A (CR)-order relation relationship differs from traditional interval-order relationships in that it is calculated as follows: . There are several advantages to using this ordered relationship, including the fact that the inequality terms deduced from it yield much more precise results than any other partial-order relation defined in the literature. This study introduces the concept of harmonical -convex functions associated with the center-radius order relations, which is very novel in literature. Applied to uncertainty, the center-radius order relation is an effective tool for studying inequalities. Our first step was to establish the Hermite−Hadamard inequality and then to establish Jensen inequality using these notions. We discuss a few exceptional cases that could have practical applications. Moreover, examples are provided to verify the applicability of the theory developed in the present study.
1. Introduction
Uncertainty problems can be distorted by using specific numbers. It is therefore crucial to avoid such errors and obtain effective results. Furthermore, Moore [1] proposed and investigated interval analysis for the first time in 1966. It is a discipline in which an uncertain variable is represented by an interval of real numbers. Through this analysis, the accuracy level of problems is improved. A variety of fields have been affected by it over the past 50 years, including differential 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].
The 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. This area of research has grown in popularity due to its diverse applications in different fields. Convexity is an integral part of optimization concepts and is widely used in operation research, economics, control theory, decision-making, and management. There 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. This 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 different integral operators, were used to establish a strong interrelationship between inequalities and interval-valued functions . 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 , 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 . A famous double inequality is defined 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 definition of convex mappings, the first 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-convex, and utilizing these notions, Jensen and inequality were established. There 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 interval-valued functions and developed these inequalities see reference [19]. Initially, Awan et al. introduced the notion of -convex functions and developed the following results [20]. Later, different authors used the notion of -convexity and developed the following inequalities using related classes of convexity see references [21–23]. The results developed using partial order relations, including inclusion relations, pseudo-order relations, and fuzzy order relations are not as accurate as the results developed using the center-radius method. Therefore, center-radius order is an ideal tool for studying inequalities, and it was developed by the following author [24]. Based on harmonical -h-convex functions, this result can be proved by Liu et al. [25].
Theorem 1 (See [25]). Consider . Define and . If and , then
The set of all harmonically --convex functions over is denoted by , 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 , . If is nonnegative super multiplicative function and then this holds:
There is novelty and significance in this study because for the first time, harmonical -convexity is connected with center-radius order relations. Furthermore, this class is more generalized since different choices of result in different classes of harmonic convex functions. There are a variety of partial order relations, but -order is distinct from them. The center and radius concepts can be calculated by using the endpoints of intervals such as: and , respectively, where .
We get our research ideas from the extensive literature and specific articles, see references [21, 25]. Using the notions of harmonical convexity and center-radius order, we introduce a novel class of convexity called harmonical --convex functions. By utilizing this new idea, we developed and Jensen-type inequalities. Furthermore, the study provides relevant examples to back up its findings.
2. Preliminaries
This section summarizes some fundamental concepts, results, and definitions. 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 arithmeticwhere . Suppose be the collection of all closed and positive intervals of , respectively. We will now talk about certain interval arithmetic algebraic properties.
Let , then and are the center and radius of interval , respectively. The center-radius form of interval can be represented as follows:
Definition 1. The CR-order relation for and can be represented as follows:
NOTE: For any arbitrary two intervals , this holds either or . Riemann integral operators for are represented as follows:
Definition 2 (See [25]). Let be an interval-valued function such that . Then, is Riemann integrable on iff are on , that is as follows:The bundle of all on is represented by .
Shi et al. [25] demonstrated that the integral retains order on the basis of CR-order relations.
Theorem 3. Let be given by and . If and , then
The following example will help to prove the abovementioned theorem.
Example 1. Let , , and for , we haveFrom Definition 1, we have and (see Figures 1–3).
SinceNow, again using the Definition 1, we have
2.1. Some Novel Definitions Pertaining to Total Order Relations
Definition 3 (See [25]). Let and be two non-negative functions. Then, is said to be a harmonically -convex function or that , if for all , and , we haveIf in (12) replaced with it is called harmonical -concave function or .
Definition 4 (See [25]). Let and be non-negative functions. Then, is said to be a harmonical -convex function, or that , if for all and , we haveIf in (13)” ” replaced with” ” it is called harmonical -concave function or .
Now let us introduce the notion of harmonically CR-convexity.
Definition 5 (See [25]). Let be non-negative interval-valued function given by and be a non-negative function. Then, is said to be a harmonical --convex function, or that , if for all and , we haveIf in (14)” ” replaced with” ” it is called harmonical --concave function or
Our next step will be to define a novel definition for harmonically --convex functions.
Definition 6. Let be non-negative interval-valued function given by and be non-negative functions. Then is said to be harmonical --convex function, or that , if for all and , we haveIf in (15)” ” replaced with” ” it is called harmonical --concave function or .
Remark 1. (1)If , Definition 6 becomes a harmonical CR-P-function [25].(2)If , , Definition 6 becomes a harmonical CR-GL-convex function [25].(3)If , , Definition 6 becomes a harmonical CR-h-convex function [25].(4)If , , Definition 6 becomes a harmonical CR-s-convex function [25].
3. Main Results
Proposition 1. Consider given by . If and are harmonical -convex over , then is a harmonical CR--convex function over .
Proof. Since and are harmonical -convex over , for each and for all , we haveNow, iffor each and for all accordinglyOtherwise, for each and for all Combining all the above, from Definition 6, it can be written asfor each and for all .
This completes the proof.
Example 2. Consider , , and . are defined as follows:Then,It is obvious that are harmonical convex functions over [0, 1] (see Figure 4). This implies that from Proposition 1, is also a harmonical CR- convex function on [0,1].
3.1. Some Variants of Hermite−Hadamard Inequalities for Harmonical-Convex Mappings Using Total Order Relations
Theorem 4. Let and . Let , if , and , we have
Proof. Since , we haveBy integrating of the above inequality over (0,1), we haveBy Definition 6, we haveBy integrating of the abovementioned inequality over (0, 1), we haveAccordinglyNow, combining (26) and (29), we get required result
Remark 2. (1)If , then Theorem 4 becomes result for harmonical CR- P-function:(2)If , , then Theorem 4 becomes result for harmonical CR-h-Godunova−Levin-function(3)If , , then Theorem 4 becomes result for harmonical CR-h-convex function(4)If , , then Theorem 4 becomes result for harmonical CR--Godunova−Levin function
Example 3. Further by Example 2, we haveAs a resultConsequently, Theorem 4 is verified.
Theorem 5. Let and . Let , if and , we havewhere
Proof. Take , we haveBy integrating of the abovementioned inequality over (0,1), we haveAccordinglySimilarly for interval , we haveAdding inequalities (41) and (42), we getNow
Example 4. Further by Example 3, we haveThus, we obtainConsequently, Theorem 5 is verified.
Theorem 6. Let such that . If , , and then, we havewhere
Proof. Consider , then, we haveThenBy integrating of the above inequality over (0,1), we haveIt follows that
Example 5. Let , , be defined as follows:ThenIt follows thatConsequently, Theorem 6 is verified.
Theorem 7. Let such that . If , , and then, we have
Proof. Since and , we haveThenBy integrating of the above inequality over (0,1), we haveDivide both sides by above equation, we get the required resultThe abovementioned theorem is proved.
Example 6. Further by Example 5, we haveIt follows thatConsequently, Theorem 7 is verified.
Next, we will establish Jensen inequality for harmonical CR--convex mapping.
4. Jensen-Type Inequality for Harmonical CR--Convex Mappings
Theorem 8 (See [25]). Let , . If is super multiplicative non-negative functions and if . Inequality then becomes as follows:where
Proof. When , then (63) holds. Suppose that (63) is also valid for , thenIt follows from mathematical induction that the conclusion is correct.
Remark 3. (1)If , Theorem 8 becomes result for harmonical CR- P-function(2)If , Theorem 8 becomes result for harmonical CR-convex function(3)If , Theorem 8 becomes result for harmonical CR-h-convex function(4)If , Theorem 8 becomes result for harmonical CR-h-GL-function(5)If , Theorem 8 becomes result for harmonical CR-s-convex function
5. Conclusions
In this article, we developed the notion of harmonically center-radius order -convex mappings. By using these notions, we developed 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 findings. These 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 different kinds of mathematical modeling. Various integral operators are appropriate for different practical problems. It is anticipated that this concept will be beneficial to other researchers working in a range of scientific disciplines.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Conceptualization was done by W.A., K.S., and M.A.; J.K.K.A. was responsible for validation; A.M.G., W.A., and K.S. performed the investigation; W.A., K.S., and M.A. wrote the original draft; J.K.K.A. and A.M.G. reviewed the manuscript; K.S was responsible for supervision; W.A. and M.A. were responsible for project administration. All authors have read and agreed to the published version of the manuscript.
Acknowledgments
This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).