New Hermite–Hadamard Inequalities for Convex Fuzzy-Number-Valued Mappings via Fuzzy Riemann Integrals

This study uses fuzzy order relations to examine Hermite–Hadamard inequalities (HHinequalities) for convex fuzzy-number-valued mappings (FNVMs). The Kulisch–Miranker order relation, which is based on interval space, is used to define this fuzzy order relation which is defined level-wise. By utilizing this idea, several novel HHand HH-Fejér-type inequalities are established in the fuzzy environment via convex FNVMs. Additional novel HH-type inequalities for the product of convex FNVMs are also found and proven with the use of practical examples. Additionally, certain unique situations that can be seen as applications of fuzzy HH-inequalities are presented. The ideas and methods presented in this work might serve as a springboard for more study in this field.


Introduction
Convex analysis has contributed significantly and fundamentally to the development of several practical and pure scientific domains. In the recent years, there has been a lot of focus on examining and separating various applications of the traditional concept of convexity. Convex mappings have recently undergone a number of expansions and generalizations. See [1][2][3][4][5][6][7] and the references therein for further helpful information. In the classical approach, a real-valued mapping U : K → R is called convex if for all x, y ∈ K, s ∈ [0, 1], where K is a convex set.
Research on the idea of convexity with integral problems is fascinating. As a result, several inequalities have been applied to convex mappings. A fascinating result of convex analysis is the Hermite-Hadamard inequality (HH-inequality, for short). The HH-inequality [8,9] for convex mapping U : K → R on an interval K = [ρ, ς] is for all ρ, ς ∈ K. Fejér considered the major generalizations of the HH-inequality in [10] which is known as the HH-Fejér inequality. Let U : K → R be a convex mapping on a convex set K and, ς ∈ K with ρ ≤ ς. Then, If C(x) = 1, then we obtain (2) from (3). Many inequalities may be found using special symmetric mapping C(x) for convex mappings with the help of inequality (3).
On the other hand, automated error analysis is performed in order to increase the accuracy of the computation results. Moore [11], Kulish and W. Miranker [12], and others conceived and studied the idea of interval analysis, which substitutes interval operation for real operations. In this field, an interval of real numbers is used to represent an uncertain variable. Based on the aforementioned literature, Zhao et al. [13] proposed hconvex interval-valued mappings in 2018 and demonstrated that the HH-inequality applies specifically to convex i.v.ms as a particular case: for all x ∈ [ρ, ς], where U * (x) is a convex mapping and U * (x) is a concave mapping. If U is Riemann integrable, then where R + I is the set of positive real intervals. We refer readers to [14][15][16][17][18][19][20][21][22][23] and the references therein for more study of the literature on the uses and characteristics of generalized convex mappings and HH-integral inequalities.
This plays a significant role in the study of a wide range of problems arising in pure mathematics and applied sciences, including operation research, computer science, management sciences, artificial intelligence, control engineering, and decision sciences. In [24], an enormous amount of research on fuzzy sets and systems has been devoted to the development of various fields. Similar to this, the concepts of convexity and non-convexity are crucial in optimization in the fuzzy domain because they allow us to characterize the optimality condition of convexity and produce fuzzy variational inequalities. As a result, the theories of variational inequality and fuzzy complementary problems have powerful mechanisms of mathematical problems and a cordial relationship. This field is fascinating and has produced many writers. Additionally, the concepts of convex fuzzy mapping and finding its optimality condition with the aid of fuzzy variational inequality were studied by Nanda and Kar [25] and Chang [26]. Fuzzy convexity's generalization and extension are crucial to its application in a variety of contexts. Let us remark that preinvex fuzzy mapping is one of the most often discussed kinds of nonconvex fuzzy mapping. This concept was first proposed by Noor [27], who also demonstrated some findings that show how fuzzy variational-like inequality distinguishes the fuzzy optimality condition of differentiable fuzzy preinvex mappings. For a more in-depth review of the literature on the uses and characteristics of generalized convex fuzzy mappings and variational-like inequalities, see [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] and the references therein.
The fuzzy mappings are fuzzy mappings with numerical values. There are certain integrals that deal with FNVMs and have FNVMs as their integrands. For instance, Osuna-Gomez et al. [43] and Costa et al. [44] built the Kulisch-Miranker order relation for the Jensen's integral inequality for FNVMs. Costa and Roman-Flores provided Minkowski and Beckenbach's inequalities, where the integrands are FNVMs, by employing the same methodology prompted by [13,43,44] and in particular by Costa et al. [45], who created a connection between the elements of fuzzy number space and interval space and developed level-wise fuzzy order relations on fuzzy number space through Kulisch-Miranker order relations defined on interval space. Using this idea of fuzzy number space, we develop a fuzzy integral inequality for convex FNVM, where the integrands are convex FNVM, Mathematics 2022, 10, 3251 3 of 18 and generalize integral inequality (2) and (3). For more information, see  and the references therein.
The structure of this study is as follows: Preliminary ideas and findings in interval space, the space of fuzzy numbers, and convex analysis are presented in Section 2. Convex FNVMs are used in Section 3 to obtain fuzzy HH-inequalities. To support our findings, some compelling instances are also provided. Conclusions and future plans are provided in Section 4.

Preliminaries
Let R be the set of real numbers and R I be the collection of all closed and bounded intervals of R, that is,  The concept of a Riemann integral for i.v.m, first introduced by Moore [33], is defined as follows: The collection of all Riemann integrable real-valued mappings and Riemann integrable i.v.m is denoted by R [c, ς] and F R [c, ς] , respectively.
Let R be the set of real numbers. A fuzzy subset set A of R is distinguished by a mapping T : R → [0, 1] , called the membership mapping. In this study, this depiction is approved. Moreover, the collection of all fuzzy subsets of R is denoted by E.
The collection of all real fuzzy numbers is denoted by E F .

Fuzzy Hermite-Hadamard Inequalities
In this section, we propose If U(x) is a concave FNVM, then (14) is reversed.
Proof. Let U : [ρ, ς] → E F be a convex FNVM. Then, by hypothesis, we have Therefore, for every ∈ [0, 1], we have It follows that Thus, In a similar way as above, we have Combining (14) and (15), we have Hence, the required result.

Remark 4.
If U * (x, ) = U * (x, ) with = 1, then Theorem 7, reduces to the result for convex mapping: We can easily note that, due to the convexity of end-point mappings, U * (x, ) and U * (x, ) with = 1 have the following possibilities to satisfy (16): either both are convex or affine convex. However, in the case of the interval HH-integral inequality (2), both end-point mappings should be affine convex because in interval inclusion U * (x) is convex and U * (x) is concave.
We now obtain some HH-inequalities for the product of convex FNVMs. These inequalities are refinements of some known inequalities, see [27,37].     Proof. By hypothesis, for each ∈ [0, 1], we have , that is, Hence, the required result.
Hence, Theorem 10 is verified. We now give HH-Fejér inequalities for convex FNVMs. Firstly, we obtain the second HH-Fejér inequality for a convex FNVM.

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After adding (20) and (21) and integrating over [0, 1], we get Since C is symmetric, then Then, from (22), we have Next, we construct the first HH-Fejér inequality for a convex FNVM, which generalizes the first HH-Fejér inequalities for convex mapping, see [10].

Conclusions
In this paper, we constructed various new HHand HH-Fejér-type inequalities for convex FNVMs, and HH-inequalities hold for this notion of convex FNVMs. We plan to investigate this idea for non-convex FNVMs and a few applications in fuzzy nonlinear programming in the future. This idea opens up a new area of research for convex analysis and optimization theory. We think that this idea will be useful to other authors as they play their roles in various scientific domains.