1. Introduction and Preliminaries
Convexity theory has been a prolific source of various inequalities found in the literature. Among these, Hadamard’s inequality is particularly well-known and widely utilized. This inequality is commonly expressed as follows:
Theorem 1 ([
1]).
Suppose is a convex function with . In that case, In recent decades, researchers have focused on generalizing convex functions to obtain novel and innovative inequalities, as demonstrated in various works [
2,
3,
4,
5,
6,
7]. Interval analysis is a methodology that deals with interval variables instead of point variables and reports computational results in the form of intervals, eliminating errors that may lead to incorrect conclusions. The first book on interval analysis was published by Moore in 1966 [
8]. In addition, Ref. [
9] provides an in-depth discussion of interval arithmetic.
We define an interval as a closed, bounded subset
O of the real numbers that take on real values. It is defined as
where
and
. The left and right endpoints of an interval
O are denoted by
and
, respectively. An interval
is considered positive if
. We use
and
to denote the sets of all closed intervals and positive closed intervals of the real numbers, respectively.
Additional results related to integral inequalities with I.V. functions can be found in [
10,
11,
12,
13,
14,
15]. However, these results rely on partial-order relations such as the inclusion relation and pseudo-order relations. In contrast, Bhunia et al. [
16] introduced a total order to an interval represented in center and radius form, as follows:
The total order or center-radius order relation between two intervals is defined as:
Definition 1 ([
16]).
For two intervals and , we define the -order relation as: Thus, for any two intervals , either or .
The concept of the Riemann integral for I.V. functions was first introduced by Moore et al. [
9]. Let
and
denote the sets of all Riemann integrable I.V. and real-valued functions on
, respectively. The following theorem establishes the relationship between
-integrable functions and Riemann integrable
-integrable functions.
Theorem 2 ([
9]).
Consider be an I.V. function, iff and The following theorem is proved by Shi et al. [
17], showing the order preservation of integrals concerning
-order.
Theorem 3 ([
10]).
Consider be two I.V. functions, . If and , then The notion of
-convex functions was introduced by Rahman et al. [
18], which led to research on the non-linear constrained optimization problems associated with this
-order. This idea opened up new avenues of research in the field of inequalities, and many researchers have obtained useful results. Shi et al. [
17], Liu et al. [
19], and Soubhagya et al. [
20] linked this
-interval order with (
h-convex,
h-harmonic convex, and
h-preinvex) I.V. functions, respectively, and derived Hermite Hadamard, Feje’r, and Pachpatte-type inequalities.
Additionally, fractional calculus studies the integrals and derivatives of functions with non-integer orders, for detail, see [
21,
22,
23]. Riemann–Lioville fractional integrals were introduced by Kilbas et al. [
22], and Lupulescu [
24] and Budak et al. [
25] introduced the I.V. left- and right-sided Riemann–Liouville fractional integrals, respectively. Many researchers have used these fractional integral operators to prove and generalize Hadamard’s inequality for different classes of convex functions; see [
26,
27,
28,
29].
Definition 2 ([
24]).
Consider be an I.V. function, . Then, the I.V. left-sided Riemann–Liouville fractional integral of function Ξ is defined by where is the Gamma function. The corresponding I.V. right-sided Riemann–Liouville fractional integral of a function by Budak et al. [
25] is as follows:
Definition 3 ([
24]).
Consider be an I.V. function, . Then the I.V. right-sided Riemann- Liouville fractional integral of the function Ξ is defined by where is the Gamma function. For recent developments regarding fractional calculus, see [
30,
31,
32,
33,
34].
In this paper, our main objective is to introduce a novel class of I.V. functions called --convex functions and explore their potential in deriving refined versions of fractional integral inequalities, such as Hermite-Hadamard’s, Fejer and Pachpatte type. We aim to demonstrate the usefulness of this new class of functions by presenting several important results that can be viewed as special cases. In addition, we will provide proof for both discrete and integral forms of Jensen’s inequalities utilizing the definition of --convex I.V. functions. To further support our claims, we will provide examples and graphical representations to validate our results.
2. On I.V. --Convex Functions and Jensen’s Type Inequalities
In this section, we define the class of --convex functions and derive discrete and integral versions of Jensen’s type inequality by utilizing this definition.
Definition 4. Let be a real function. Let ; then, Ξ is said to be γ-convex function iffor all and . If the function
satisfies the following inequality
for all
, then
is said to be super-multiplicative. If the sign in inequality (
2) is replaced by ≤, then
is said to be sub-multiplicative.
Definition 5. Let be a real function. Let be an I.V. function such that ; then, is said to be I.V--γ-convex function iffor all and . Throughout the paper, we will denote the set of all
-
-convex functions on
by
Remark 1. If we put in Definition 5, then it reduces to the classical definition of γ-convex functions.
Remark 2. If in (3), then we have the definition of -convex functions [18]. In [
17], we can see the following classes deduced from inequality (
3) as:
When
, then we have the definition of
-
P functions.
When
,
, then we have the definition of
-
-convex functions.
To the best of our knowledge, the following are the new classes as special cases of inequality (
3):
When
,
, then we have the definition of
-
-Godunova–Levin functions.
When
, then we have the definition of
-tgs-convex functions.
Proposition 1. Consider such that and with . If then .
Proof. Since
, then for all
and
, we have
Hence, . This completes the proof. □
Proposition 2. Let be an I.V. function, such that . If and are γ-convex functions over , then is said to be -γ-convex function over .
Proof. It is given that
and
are
-convex functions over
; then, for each
and
, we have
If
then, for each
and
, we have
which implies
Otherwise, for each
and
, we have
implies that
From Definition 5, we can write
for each
and
. This completes the proof. □
Example 1. Let , and be defined by , then and .
Obviously, and are γ-convex functions on . From Proposition 2, is -γ-convex function on
We now give a visual explanation of Example 1 in
Figure 1.
We move towards the proof of discrete Jensen inequality for --convex functions.
Theorem 4. Let be a real function and let be an I.V. function, such that , and . If γ is a non-negative super multiplicative function and , thenwhere . Proof. Suppose that
. For
(
3) implies
Suppose that (
4) holds for
; then,
We prove the inequality (
4) holds for
; then,
From the definition of
-
-convexity of the function
, we have
and
This completes the proof. □
Remark 3. If we use in Theorem 4, then it reduces to the classical case.
Remark 4. In [17], the authors have proved the following classes of discrete Jensen’s inequality (4): When , then we can obtain a Jensen inequality for -convex functions. When , then we can obtain a Jensen inequality for -P-convex functions. When , , then we can obtain a Jensen inequality for -γ--convex functions.
To the best of our knowledge, the following are the new classes of discrete Jensen’s inequality (
4).
When
,
, then we can obtain a Jensen inequality for
-
-Godunova-Levin functions.
When
, then we can obtain a Jensen inequality for
-
-convex functions.
Letting in Theorem 4, we can obtain the following result:
Corollary 1. Let be a real function and let be an I.V. function such that , . If γ is a non-negative super multiplicative function and , then Theorem 5. Let be a real function and let be an I.V. function such that . If γ is a non-negative super multiplicative function and , then for , such that , we have Proof. Let
. Let
and
, then by given assumptions, we have
implies that
and
Now, substituting
, then from (
3), we have
Combining (
6) and (
7), we have
This completes the proof. □
We establish a refinement of discrete Jensen inequality for --convex functions.
Theorem 6. Let be a real function and let be an I.V. function such that , and . If γ is a non-negative super multiplicative function, and , thenwhere . Proof. Letting
in (
8) for the function
, we have
Multiplying the above inequalities by
and also for
and adding them, we have
This completes the proof. □
Remark 5. If we use in Theorem 6, then it reduces to the classical case.
Theorem 7. Let , and be a super multiplicative function. Let be an I.V. function, such that and . Ifexists and finite, then Proof. Let
be a partition given by
where
for
. By choosing an arbitrary point
, then, using the definition of Riemann sum
Since
is
-
- convex and continuous,
, then the composition function
and we have
Taking the limit
on both sides, we have
Since and
This completes the proof. □
Remark 6. If , then Theorem 7 reduces to the classical case.
3. Main Results
In this section, we will derive Hadamard, Pachpatte and Fejér-type integral inequalities for --convex I.V. functions.
Theorem 8. Let be a real function, . Let be an I.V. function, such that and . If ; then, Proof. Using the Definition of I.V.
-
-convex function, we have
Substituting
, multiplying both sides by
and integrating over
, we have
After applying Theorem 2 to both sides and simple calculations, we have
For the proof of other inequalities, using the definition of I.V.
-
-convex function and multiplying both sides by
and integrating over
, we have
From (
10) and (
11), we have
This completes the proof. □
Remark 7. If we take and in Theorem 8, then it is identical to Theorem 4.1 for in [17]. Remark 8. If we set , and in Theorem 8, then it reduces to the classical inequality (1). From Theorem 8, we can deduce the following special cases:
When
, then we have Hadamard’s inequality for
-convex functions.
When
, then we have Hadamard’s inequality for
-
P functions.
When
,
, then we have Hadamard’s inequality for
-
-convex functions.
When
,
, then we have Hadamard’s inequality for
-
-Godunova-Levin functions.
When
, then we have Hadamard’s inequality for
-
-convex functions.
Theorem 9. Let be a real function. Let be two I.V. functions such that , and . If and ; then,where Proof. Since
and
and
; then,
Adding (
13) and (
14), and multiplying both sides by
and integrating over
, we have
This completes the proof. □
Remark 9. If we take and in Theorem 9, then it is identical to Theorem 4.7 for in [17]. Remark 10. If we use and in Theorem 9, then it reduces to the classical case.
From Theorem 9, we can deduce the following special cases:
When
, then we have the result for
-convex functions.
When
, then we have the result for
-
P functions.
When
,
, then we have the result for
convex functions.
When
,
, then we have the result for
-
-Godunova-Levin functions.
When
, then we have the result for
-
- convex functions.
Theorem 10. Let be a real function, . Let be two I.V. functions such that , and . If and , then
where
and
are defined in Theorem 9.
Proof. Since
and
, we have
Using (
17) in (
16) and multiplying both sides by
and integrating over [0, 1], we obtain
This completes the proof. □
Remark 11. If we take and in Theorem 10, then it is identical to Theorem 4.9 for in [17]. Remark 12. If we put and in Theorem 10, then it reduces to the classical case.
From Theorem 10, we can deduce the following special cases:
When
, then we have the result for
-convex functions.
When
, then we have the result for
-
P functions.
When
,
, then we have the result for
-
convex functions.
When
,
, then we have the result for
-
-Godunova-Levin functions.
When
, then we have the result for
-tgs- convex functions.
The following result is fejér-type inequality for I.V. --convex functions.
Theorem 11. Let be a real function, . Let be an I.V. function such that , and and symmetric with respect to . If , then Proof. Using the definition of I.V.
-
-convex function and multiplying both sides by
and integrating over
, we have
Therefore, (
19) implies that
For the proof of other side inequality, using the definition of I.V.
-
-convex function and multiplying both sides by
, we can obtain
Multiplying both sides by
and integrating over
, we have
From (
20) and (
21), we have
□
Remark 13. If we use and in Theorem 11, then it reduces to the classical case.
From Theorem 11, we can deduce the following special cases:
When
, then we can obtain the fejér inequality for
-convex functions.
When
, then we can obtain the fejér inequality for
-
P functions.
When
,
, then we obtain fejér inequality for
-
-convex functions.
When
,
, then we can obtain the fejér inequality for
-
-Godunova-Levin functions.
When
, then we can obtain the fejér inequality for
-
-convex functions.
4. Numerical Examples and Graphical Explanation
In this section, we will verify our main results with the help of numerical examples and a graphical demonstration.
Example 2. Let , then Theorem 8 implies that For the purpose of visually representing Theorem 8, we will use the same assumptions as used in the numerical calculation. However, in this case, we will set . Theorem 8 is satisfied.
Figure 2 presents visual analysis of Theorem 8.
Example 3. Let , then Theorem 9 implies that For the purpose of visually representing Theorem 9, we will use the same assumptions as used in the numerical calculation. However, in this case, we will set , then Theorem 9 is satisfied.
Figure 3 presents visual analysis of Theorem 9.
Example 4. Let ; then, Theorem 10 implies that For the purpose of visually representing Theorem 10, we will use the same assumptions as used in the numerical calculation. However, in this case, we will set , then Theorem 10 is satisfied.
Figure 4 presents visual analysis of Theorem 10.
Example 5. Let , for and for , then Theorem 11 implies that For the purpose of visually representing Theorem 11, we will use the same assumptions as used in the numerical calculation. However, in this case, we will set , then Theorem 11 is satisfied.
Figure 5 presents visual analysis of Theorem 11.
5. Applications to Special Means
For positive real numbers and , the following means are well known:
Proposition 3. Let and ; then, the following inequality holds: Proof. By taking
and
in Theorem 8, we have
then
□
6. Conclusions
Over the years, classical integrals inequalities have been generalized by different innovative and novel techniques, such as by employing fractional calculus, quantum calculus, and fuzzy concepts. A significant amount of fractional variants of Hermite–Hadamard-type inequalities are present. For the sake of the unification of trapezium-type inclusions, we organized the current investigation. In the current study, we introduced a new class of interval-valued convexity named -, involving a non-negative mapping and ordering relation, which is the total order relation. We discussed some fundamental integral inequalities as applications. Initially, we computed new generalizations of Jensen’s type containments and their consequences. By making use of this generic class and fractional concepts, we proposed new fractional analogs of Hermite–Hadamard inequality; its weighted form is associated with a symmetric function at about the mid-point and Pachpatte’s type inclusions for the product of two I.V - convex functions. We also verified our primary findings via numeric examples and graphical illustrations. In future, we will study this class from the perspective of the fuzzy domain and, hopefully, this class will play a vital role in the theory of inequalities and other related problems in optimization.
Author Contributions
Conceptualization, S.R. and M.U.A.; methodology, M.V.-C., S.R., M.U.A., M.Z.J., A.G.K. and M.A.N.; software, M.V.-C., S.R., M.U.A. and M.Z.J.; validation, M.V.-C., S.R., M.U.A., M.Z.J., A.G.K. and M.A.N.; formal analysis, M.V.-C., S.R., M.U.A., M.Z.J., A.G.K. and M.A.N.; investigation, M.V.-C., S.R., M.U.A., M.Z.J., A.G.K. and M.A.N.; writing—original draft preparation, M.V.-C., S.R., M.U.A., M.Z.J., A.G.K. and M.A.N.; writing—review and editing, M.V.-C., S.R., M.U.A., M.Z.J. and A.G.K.; visualization, S.R., M.U.A., M.Z.J. and M.A.N.; supervision, M.U.A. All authors have read and agreed to the published version of the manuscript.
Funding
This research is supported by Pontificia Universidad Católica del Ecuador Proyect Tí- tulo: “Algunos resultados Cualitativos sobre Ecuaciones diferenciales fraccionales y desigualdades integrales” Cod UIO2022.
Acknowledgments
The authors express gratitude to the editor and anonymous reviewers for their valuable comments and suggestions.
Conflicts of Interest
The authors declare no conflict of interest.
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