1. Introduction and Preliminaries
Convex mapping theory is a bridge between geometrical and analytical properties and modernizes various concepts. Also, it provides unified approaches to assessing the various problems of optimization, operational research, differential equations, and inequalities. Observing the dominance of convexity, various new concepts of convex sets and mappings have been explored, relying on weighted means and more general frameworks like interval-valued calculus, fuzzy-valued calculus, and stochastic analysis. The fundamental goal of such a theory is to define new generalizations of convexity and its characterization through inequalities. For more details, see [
1,
2,
3,
4,
5].
The role of inequalities is fundamental in the analysis of complex mathematical expressions that appear in most branches of the physical sciences. Their work has grown out of multiple viewpoints: the extension of applicability and overcoming the shortcomings of classical results, the use of tools of functional analysis, further developments in calculus, and convexity. The inequalities were originally devised to bound mappings and integrals but have evolved into essential tools for the uniqueness and stability of solutions, for estimating errors in numerical integration, and for dealing with problems in information theory, among other things. Convex analysis has substantially enriched this field, because many classical inequalities arise naturally from properties of convex functions. For more information, see [
6,
7].
Let us mention the premises of convex mappings.
Definition 1 ([
8])
. Any mapping is said to be convex if To highlight the strong enhancement between inequalities and convexity, we recover some fundamental results. First, we furnish Jensen’s inequality:
Theorem 1 ([
8])
. If is a convex mapping, , and then Likewise, another attracting double inequality due to convex mapping is Hermite–Hadamard inequality and is given as
Theorem 2 ([
8])
. If is a convex mapping then for , as well as , the inequality Set-valued analysis is an effective and attractive branch of mathematics; it generalizes single-valued mappings. Interval-valued and fuzzy-valued are specific cases of it. These kinds of mappings are quite applicable to approximating the error analysis of numerical methods. The subject of interval analysis was pioneered by Moore [
9], and monographs authored by him are regarded as fundamental to exploring this subject further. In recent times, it has influenced every aspect of mathematical sciences because a natural problem is under what condition single-valued results can be updated for set-valued maps. In the following context, Breckner [
10] proved the Hermite–Hadamard leveraging the set-valued convexity, which is given as follows:
Let
be a set-valued convex mapping; then,
In 1965, Zadeh [
11] generalized the idea of characteristic mapping and defined membership mapping to deal with quantities having vagueness. He delivered the premises of the convex set and mapping in a fuzzy context. Moreover, the concept of convexity has a fundamental role in defining fuzzy numbers. These concepts provide a new strategy of estimating quantities and models with certain attributes other than statistical approaches. Zadeh’s contributions are fundamental in the progression of possibility theory; they are marvelous contributions to exploring the problems of computer science and dynamical systems. In 1982, Dubois and Prade [
12] worked on fuzzy calculus and explored the preliminary concepts. In 1992, Nanda and Kar [
13] studied the numerous classes of convexity, including their essential attributes.
In order to prove our main result, let us mention the facts and rules regarding interval and fuzzy analysis. Throughout the investigation, we denote the space compact intervals and non-negative compact intervals by
and
, respectively. Let
, and
; then, Minkowski’s operations are given as
Then, the Minkowski difference
, addition
, and
for
are defined by
and
Definition 2 ([
14])
. The "" relation over is demonstrated as if and only if for all is a pseudo-order or left–right (LR) ordering relation. The ranking relation defined based on center and radius of interval is reported as follows.
Definition 3 ([
14])
. The relation between two intervals and is defined as Thus, for any two intervals , either or .
Theorem 3 ([
14,
15])
. Consider and fuzzy set ξ. Its ρ-level is defined as of , and .For greater conciseness, we symbolize the space of all real fuzzy sets and fuzzy numbers by and , respectively. If a fuzzy set meets the conditions, such as normality, semi-continuity, fuzzy convexity, and being compactly supported, then it is called fuzzy number (interval).
Let be a fuzzy interval ⇔-levels is a compact convex set of .
Its
-level form is given as
where
Also, fuzzy numbers can be studied and explored in triplet, depending on the parameters, as follows:
The aforementioned representation is more instrumental for studying fuzzy numbers in different fields. To become more familiar with these concepts, consult [
16].
Proposition 1. If then the relation “≼" expressed on byThe following relation satisfies the properties of partial order relation. For and the scalar product sum with constant, the sum and product of two fuzzy numbers, are expressed by Definition 4 ([
17])
. A fuzzy mapping is fuzzy number-valued. If then ρ-cuts showcase the cluster of I.V.F, such that are given by for all . Also, left and right real-valued mappings are termed as end-point mappings of . Definition 5 ([
18])
. Presume that is an F.I.V mapping. Then, the fuzzy Riemann definite integral of over , represented by , for all , where describes the space of integrable mappings. Definition 6 ([
13])
. The fuzzy numbered mapping is termed as a convex mapping on if for all . Over the years, fractional calculus has forced researchers to focus on the transformation of existing mathematical models to general concepts. Fractional derivatives and integral operators are used to showcase the various physical quantities with more precision. Although these operators have few limitations, they generalize classical operators, and by utilizing fractional operators one can attain the analytical solution for those differential systems whose analytical solutions do not exist or are difficult to calculate. Recently, mathematicians have focused on an approach combining fractional and fuzzy calculus to study differential systems. Let us reformulate the notion of fuzzy Riemann–Liouville fractional operators relying on -levels.
Definition 7 ([
19])
. Let , and contains the collection of fuzzy-valued Lebesgue measurable mapping on . Then, and Furthermore, the left and right RL-fractional operator based on left and right end-point mappings can be defined; that is, where and By a similar argument, we can define the right operator. In [
20], the authors constructed the various kinds of inequalities in association with approximately interval-valued convexity. Nwaeze et al. [
21] discussed interval-valued
n-polynomials to establish new generalized inequalities. In [
22], the authors presented unified
convexity to analyze new counterparts of existing results of inequalities. Liu et al. [
23,
24] formulated some novel consequences of totally ordered
and harmonic convexity. Budak and his co-authors [
25] employed classical interval-valued convexity RL operators to explore fractional analogs for the first time within an interval-valued setting. In [
26], the authors focused on generalizing the existing classes of convexity and delivered the conception of
totally ordered convexity along with essential inequalities. The authors of [
27] studied fractional coordinated harmonic inequalities through unified fractional operators. For deep information, read [
28,
29,
30,
31,
32]. Fahad et al. [
33] discussed interval-valued
-
convexity and associated inequalities with application to divergence measures. Javed et al. [
34] worked on stochastic generalized convexity leveraging the idea of quasi mean and looked at various kinds of inequalities. In [
35,
36], the authors transformed superquadraticity to interval-valued via partial and total ranking relation. Also, they furnished the analysis of both classes through an inequality perspective. In 1998, Furukawa [
37] discussed the Lipschitz continuity of convex fuzzy number-valued mappings. Costa et al. [
38] employed fuzzy number-valued mappings to approximate the bounds of Ostrowski’s inequality. Gong and Hai [
39] transformed the idea of vector-valued fuzzy mappings to
n-dimensional fuzzy numbers and discussed various important results. Zhang et al. [
14] initiated the LR set and fuzzy-valued mappings and proved a few Jensen’s inequalities for Aumann integrals. Cheng and his colleagues [
40] delivered fuzzy variants of trapezium-like inequalities. Khan et al. [
41,
42] bridged the concepts of fuzzy-valued convexity and RL operators to deduce new general bounds of fuzzy integrals and utilized the weighted
p-mean fuzzy ordered relation to introduce a new class of fuzzy-valued convexity and deliver some interesting inequalities. Abbaszadeh and Eshaghi [
43] developed the idea of fuzzy number-valued
r convexity to examine inequalities. Allahviranloo et al. [
44] presented the idea of extended fuzzy-valued mappings along with applications to inequalities. Bin-Mohsin [
45] put forward the idea of fuzzy number-valued bi-convexity through a bi-mapping and characterized it through classical inequalities. For a deeper understanding, read [
46,
47,
48,
49].
From the previous studies, we observe that all the classes of fuzzy-valued convex mappings are constructed by means of various partial ordering relations. An interesting question comes to mind: is it possible to define fuzzy-valued convexity utilizing total order relation? The principle of intent is to define fuzzy number-valued mapping by the ordering relation. First, we will define the class of fuzzy number-valued cr convex mappings. To extend it for a larger space of mappings, we will introduce fuzzy number-valued extended convex mappings via non-negative mapping. We will furnish some basic properties, criteria, examples, and characterizations through fractional integral inequalities. Lastly, some simulations and applications of the obtained results via two kinds of fuzzy number will be derived. We hope this class will be beneficial for studying further inequalities.
3. On Fuzzy Number-Valued Extended -ℏ-Convex Mappings
Definition 9. Let be a fuzzy number-valued mapping, such that . Then, is known as a fuzzy number-valued -ℏ convex mapping iffor all , and . Now, we enlist some potential totally ordered subclasses of Definition 9. We deduce the classes of fuzzy number-valued
convex mapping,
-
P mapping,
-
s convex mapping,
-
s-Godunova–Levin mapping, and
-tgs convex mapping by taking
in (
1), respectively.
Remark 1. Through the utilization of different values of ℏ, we obtain the various classes of totally ordered fuzzy number-valued convex mappings, like exponential convexity, n-polynomial convexity, trigonometric convexity, and many more. Assuming that in Definition 9, then we recover the class of generalized ℏ convexity.
The space of all generalized
ℏ convex mappings and the space of all fuzzy number-valued extended
-
ℏ convex mappings are represented by
and
respectively.
Proposition 3. Consider , such that with .
If then .
Proof. Since
, then for all
and
we have
Hence,
. □
Proposition 4. Let be a fuzzy number-valued mapping, such that . If then .
Proof. If
then for
and
we obtain
and
If
and
then by the definition of
-ordering relation we obtain
Otherwise, we have
Again, considering the definition of
relation, we obtain
This completes the proof. □
Now, we give some illustrative examples in favor of the newly developed class of convexity.
Example 1. Let be a triangular fuzzy number, and its membership mapping is defined as follows:Moreover, the ρ-level representation is . Also, and . Note that both and are generalized ℏ convex. This implies that is a -ℏ fuzzy-valued convex mapping. Now, we deliver the graphical depiction of triangular fuzzy number-valued mapping for
(
Figure 1).
Example 2. Let be a trapezoidal fuzzy number, and its membership mapping is defined as follows:Moreover, the ρ-level representation is . Also, and . Note that both and are convex. This implies that is a -ℏ fuzzy-valued convex mapping. Now, we deliver the graphical depiction of trapezoidal fuzzy number-valued mapping for
(
Figure 2).
3.1. Jensen’s-Type Inequalities
This section contains some celebrated Jensen’s-like inequalities. First, we prove Jensen’s inequality for .
Theorem 4. If and ℏ is supermultiplicative mapping thenwhere . Proof. Suppose that
. For
, we obtain
Consider (
2) satisfies for
; then,
This inequality can be transformed as
Now, we prove (
2) for
:
From the definition of the fuzzy number-valued extended
-
ℏ convexity of the mapping
and utilizing the supermultiplicative property of
ℏ, we have
and
Finally, we obtain
Hence, totally ordered Jensen’s inequality is acquired. □
Here are some useful deductions of fuzzy number-valued Jensen’s inequality (
2):
Substituting
in (
2) produces Jensen inequality for fuzzy number-valued
convex mappings.
To obtain fuzzy-valued Jensen’s inequality for
-
P convex mappings, we take
in (
2):
For
, the inequality (
2) generates the following inequality:
We obtain totally ordered fuzzy-valued Jensen’s inequality for
-Godunova–Levin convex mapping by substituting
in (
2):
Setting
in inequality (
2), we obtain another unified Jensen’s inequality.
For , we prove another Jensen’s inequality.
Theorem 5. Let be a supermultiplicative mapping. If and for , , such that , then Proof. Let
. Let
and
; then,
and
Selecting
, then from (
1) we obtain
Similarly,
From (
4) and (
5), we have
From the above inequality, we deduce our desired Jensen’s-like inequality. □
By incorporating the Jenesn’s-type inequality (
3), we prove another refinement of it.
Theorem 6. Let , and . If and thenwhere . Proof. The result is obvious. □
For
, we construct the continuous analogue of inequality (
2).
Theorem 7. Let be an integrable mapping, and . Ifexists and is finite then Proof. Let
be the collection of all partitions of
, and
is given by
where
for
. By selecting
, then, utilizing the Riemann sum,
Since
, then
Since
and, continuous,
, then the composition mapping
and we have
Applying limit
, we obtain
Since
and
, this completes the proof. □
3.2. Hermite–Hadamard’s Inequalities
In the following segment, we investigate trapezium-type inequalities via fuzzy number-valued extended -h convex mappings. First, we aim to develop the fractional fuzzy number-valued trapezoid inequality.
Theorem 8. Let , and is a fuzzy-valued Riemann integrable mapping. Then, for we obtain Proof. Using the definition of fuzzy number-valued extended
-
ℏ convex mapping, we have
Inserting
, taking the product on both sides by
, and integrating over
, we have
This above inequality can be written as
This implies that
Some simple computations produce the left Hermite–Hadamard’s inequality.
Again, considering the notion of fuzzy number-valued extended
-
ℏ convex mappings, we have the following inequality:
This implies
Hence, the desired right-Hadamard’s inequality is achieved. □
Theorem 8 reduces to the following potential inequalities:
Taking
in Theorem 8 yields the trapezium inequality for fuzzy number-valued
convex mappings:
For
in Theorem 8, we obtain following double inequality.
Selecting
in Theorem 8 produces the following fuzzy-valued inequality:
To generate the double fuzzy-valued inequality associated
-
convex mappings, we take
in Theorem 8:
Example 3. Let be a triangular fuzzy number-valued extended convex mapping, as defined in Example 1. Since satisfies all the credentials of Theorem 8, we have the following graphical representations for and .
To construct Figure 3a,b, we consider the triangular fuzzy number-valued extended -ℏ convex mappings, and specific values ρ. Both figures affirm the accuracy of the inequality proved in Theorem 8. Let be a trapezoidal fuzzy number-valued extended convex mapping, as defined in Example 2. Since satisfies all the credentials of Theorem 8, we have the following graphical representations for and .
To construct Figure 4a,b, we consider the trapezoidal fuzzy number-valued extended -ℏ convex mappings, and specific values ρ. Both figures highlight the accuracy of inequality proved in Theorem 8. Next, we build the weighted Hermite–Hadamrad’s inequality, based on the newly proposed class of convexity and symmetric mappings about the mid-point.
Theorem 9. Let , let be a fuzzy-valued Riemann integrable mapping, and let be a symmetric with respect to ; then, for , Proof. Using the definition of fuzzy number-valued extended
-
ℏ convex mapping and taking the product on both sides by
and integrating over
,
Then,
Also,
Finally, by comparing (6), (7), and (8), we obtain the left Fejer inequality:
For the second fuzzy number-valued Fejer inequality, again considering fuzzy number-valued extended
-
ℏ convex mapping and taking the product of the resultant inequality by
, we obtain
Taking the product of
with the aforementioned inequality, then applying integration, we have
This yields
Hence, the intended inequality is attained. □
Under certain constraints, Theorem 9 reduces to the following significant inequalities:
Inserting
into Theorem 9 produces the following inequality:
Example 4. Let be a triangular fuzzy number-valued extended convex mapping, as defined in Example 1, and let be a symmetric mapping about 1. Since and satisfy all the credentials of Theorem 9, we have the following graphical representations for and .
To construct Figure 5a,b, we consider the triangular fuzzy number-valued extended -ℏ convex mappings, and specific values ρ. Both figures affirm the accuracy of inequality proved in Theorem 9. Let be a trapezoidal fuzzy number-valued extended convex mapping, as defined in Example 2, and let be a symmetric mapping about 1. Since and satisfy all the credentials of Theorem 9, we have the following graphical representations for and .
To construct Figure 6a,b, we consider the trapezoidal fuzzy number-valued extended -ℏ convex mappings, and specific values ρ. Both figures highlight the accuracy of the inequality proved in Theorem 9. Now, we develop new bounds for the product of two generalized convex mappings.
Theorem 10. Let be two non-negative mappings. Ifand thenwhere and are defined in Theorem 11. Proof. Since
and
, we have
This results in
Also, by the definition of fuzzy number-valued extended
convex mappings, we have
From the previous inequality, we have
Finally, some simple computations deliver the desired inequality. □
Example 5. Let be two triangular fuzzy number-valued extended convex mappings, is defined in Example 1, and
Since and , all the credentials of Theorem 10 are satisfied. We have the following graphical representations for and .
To construct Figure 7a,b, we consider the triangular fuzzy number-valued extended -ℏ convex mappings, and specific values ρ. Both figures affirm the accuracy of the inequality proved in Theorem 10. Let be a trapezoidal fuzzy number-valued extended convex mapping, such thatSince and , all the credentials of Theorem 10 are satisfied. We have the following graphical representations for and . To construct Figure 8a,b, we consider the trapezoidal fuzzy number-valued extended -ℏ convex mappings, and specific values ρ. Both figures highlight the accuracy of the inequality proved in Theorem 9. Lastly, we construct the right-product inequality.
Theorem 11. Suppose are two mappings. If and thenwhere Proof. Since
and
, then
Likewise,
Summing (9) and (10), taking product by
, and integrating over
, we have
This implies that
Final inequality is achieved. □
Example 6. Let be two triangular fuzzy number-valued extended convex mappings, as defined in Example 5. Since and satisfy all the credentials of Theorem 11, we have the following graphical representations for and .
To construct Figure 9a,b, we consider the triangular fuzzy number-valued extended -ℏ convex mappings, and specific values ρ. Both figures affirm the accuracy of the inequality proved in Theorem 11. Let be a trapezoidal fuzzy number-valued extended convex mapping, as defined in Example 5. Since and satisfy all the credentials of Theorem 11, we have the following graphical representations for and .
To construct Figure 10a,b, we consider the trapezoidal fuzzy number-valued extended -ℏ convex mappings, and specific values ρ. Both figures highlight the accuracy of the inequality proved in Theorem 11. Remark 2. It is interesting to note that by utilizing the values of ℏ in Theorems 4–11 we obtain the various classes of totally ordered fuzzy number-valued convex mappings, like s convexity in all four senses, Godunova–Levin convexity, tgs convexity, exponential convexity, n-polynomial convexity, trigonometric convexity, and many more. Assuming that in Definition 9, then we recover the class of generalized ℏ convexity. Moreover, for and in Theorems 4–11 we obtain the Jensen,’s, Schur’s, Hermite–Hadamard’s, Fejer, and left- and right-product Hermite-Hadamard inequality for fuzzy number-valued convex mappings. All these results are new and novel.