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Article

# Some New Integral Inequalities for Generalized Preinvex Functions in Interval-Valued Settings

1
2
Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico
3
Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia
4
Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
*
Authors to whom correspondence should be addressed.
Axioms 2022, 11(11), 622; https://doi.org/10.3390/axioms11110622
Received: 6 October 2022 / Revised: 25 October 2022 / Accepted: 1 November 2022 / Published: 7 November 2022
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)

## Abstract

:
In recent years, there has been a significant amount of research on the extension of convex functions which are known as preinvex functions. In this paper, we have used this approach to generalize the preinvex interval-valued function in terms of -preinvex interval-valued functions because of its extraordinary applications in both pure and applied mathematics. The idea of -preinvex interval-valued functions is explained in this work. By using the Riemann integral operator, we obtain Hermite-Hadamard and Fejér-type inequalities for -preinvex interval-valued functions. To discuss the validity of our main results, we provide non-trivial examples. Some exceptional cases have been discussed that can be seen as applications of main outcomes.
MSC:
26A33; 26A51; 26D10

## 1. Introduction

Set-valued analysis, which is the study of sets in the context of mathematics and general topology, is a subset of interval analysis. A well-known example of interval enclosure is the Archimedean method, which includes calculating a circle’s circumference. The interval uncertainty that is present in many computational and mathematical models of deterministic real-world systems is addressed by this theory. This method studies interval variables rather than point variables and expresses computing results as intervals, avoiding inaccuracies that lead to inaccurate conclusions. One of the initial objectives of the interval-valued analysis was to take into account the error estimates of the numerical solutions for finite state machines. One of the essential methods in numerical analysis is interval analysis, which Moore initially introduced in his well-known work [1]. Due to this, it has found use in a variety of areas, including computer graphics [2], differential equations for intervals [3], neural network output optimization [4], automatic error analysis [5], and many more. We recommend [6,7,8,9,10,11,12,13,14,15,16,17] to readers who are interested in results and applications.
Particularly those associated with the Jensen, Ostrowski, Hermite-Hadamard, Bullen, Simpson, and Opial inequalities have a considerable impact on mathematics. Some renowned scholars have lately extended many of these inequalities to interval-valued functions (I∙V∙Fs) (see, for instance, [18,19,20,21,22,23,24,25,26,27,28]), and many have also studied the Hermite-Hadamard inequality (𝐻∙𝐻-inequality) for convex functions. The 𝐻∙𝐻-inequality for convex mapping $G : O → ℝ$ on an interval is all
On the other hand, a powerful tool for solving a wide range of problems in applied analysis and nonlinear analysis, including many concerns in mathematical physics, is the generalized convexity of mappings. Recently, extensive study has been done on a number of generalizations of convex functions. It is intriguing to explore integral inequalities from a mathematical analytic perspective. Inequalities and other extended convex mappings have been thought to be related to the study of differential and integral equations. They have made a significant contribution to a number of fields, including electrical networks, symmetry analysis, operations research, finance, decision-making, numerical analysis, and equilibrium, see [39,40,41,42,43,44,45,46,47,48,49]. We explore the possibility of encouraging the subjective features of convexity by employing a variety of basic integral inequalities.
Several types of convexity are related to the Hermite-Hadamard inequality; for several instances, see [50,51,52,53,54,55,56,57,58,59,60,61,62]. The concept of harmonic convexity and several associated Hermite Hadamard type inequalities were introduced by Iscan [63] in 2014, and 2015 saw the first description of harmonic $𝒽$-convex functions and certain related Hermite-Hadamard inequalities by the authors of [64]. In recent years, numerous studies have explored the relationship between integral inequalities and interval-valued functions, yielding many important results. Using the extended Hukuhara derivative, Chalco Cano [65] researched the Ostrowski-type inequalities, while Roman-Flores [66] established the Minkowski type inequalities and the Beckenbach type inequalities. Costa [67] introduced the Opial-type inequalities. Zhao et al. [68] recently built on this notion by incorporating interval-valued coordinated convex functions and generating related 𝐻∙𝐻 type inequalities. It was also used to support the 𝐻∙𝐻- and Fejér-type inequalities for the preinvex function [69,70] and convex interval-valued function for n-polynomials [71]. The idea of interval-valued preinvex functions, first proposed by Lai et al. [72], has recently been extended to include interval-valued coordinated preinvex functions. The 𝐻∙𝐻 inequality was expanded to include interval $𝒽$-convex functions [73], interval harmonic $𝒽$-convex functions [74], interval ($𝒽 1$, $𝒽 2$)-convex functions [75], interval ($𝒽 1$, $𝒽 2$)-convex functions, and interval harmonically ($𝒽 1$, $𝒽 2$)-convex functions [76], when interval analysis was combined. The authors in [77] used the definition of the $𝒽$-Godunova-Levin function to account for this inequality. Additionally, the authors of [78] developed a Jensen-type inequality for ($𝒽 1$, $𝒽 2$) interval-nonconvex functions, whereas the author of [78] published a fuzzy Jensen-type integral inequality for fuzzy interval-valued functions. For more information, see [79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98] and the references therein.
Our investigation was inspired by the substantial body of literature and the targeted studies [76,77]. Interval-valued ($£ 1$, $£ 2$)-preinvex functions are introduced, and new 𝐻∙𝐻-type inequalities are constructed for the previously covered topic. The following is how the paper is set up: Section 2 provides the introduction and the mathematical context. The scenario and our key findings are covered in Section 3. Section 4 contains the conclusion and future scope.

## 2. Preliminaries

Let $E C$ be the collection of all closed and bounded intervals of $ℝ$ that is If $Z * ≥ 0$, then is named as a positive interval. The set of all positive intervals is denoted by $E C +$ and defined as
If and $s ∈ ℝ$, then arithmetic operations are defined by
For the inclusion $“ ⊆ ”$ is defined by
Theorem 1
([1]). If is an 𝘐-𝘝-𝘍 such that , then $G$ is Riemann integrable over $[ ς , q ]$ if and only if, $G * ( ω )$ and $G * ( ω )$ are both Riemann integrable over $[ ς , q ]$ such that
where .
The collection of all Riemann integrable real valued functions and Riemann integrable 𝘐-𝘝-𝘍s is denoted by $ℛ [ ς , q ]$ and $T ℛ [ ς , q ] ,$ respectively.
Definition 1
([84]). Let $O$ be an invex set. Then, I∙V∙F $G : O → E C$ is said to be preinvex on $O$ with respect to $w$ if
for all  $w : O × O → ℝ .$ If $G$ is preincave IVF, then $− G$ is preinvex IVF.
Definition 2
([83]). Let $O$ be an invex set and such that $£ ≢ 0$. Then, I∙V∙F $G : O → E C$ is said to be $£$ -preinvex on $O$ with respect to $w$ if
for all where $G ( x ) ≥ 0$ and $w : O × O → ℝ .$
Definition 3
([84]). Let $O$ be an invex set and such that . Then, I∙V∙F $G : O → E C$ is said to be:
• -preinvex on$O$with respect to$w$if
for all, where$G ( x ) ≥ 0$and$w : O × O → ℝ .$
• -preincave on$O$with respect to$w$if inequality (13) is reversed.
Remark 1.
If$£ 2 ( o ) ≡ 1$, then-preinvex I∙V∙F, we acquire the definition of$£ 1$-preinvex I∙V∙F.
If $£ 1 ( o ) = o s$, $£ 2 ( o ) ≡ 1$, then from the definition of -preinvex I∙V∙F becomes $s$-preinvex I∙V∙F in the second sense, that is
If $£ 1 ( o ) = o , £ 2 ( o ) ≡ 1$, then -preinvex I∙V∙F becomes preinvex I∙V∙F.
If $£ 1 ( o ) = £ 2 ( o ) ≡ 1$, then from the definition of -preinvex I∙V∙F, we acquire the definition of $P$ I∙V∙F, that is, see [83]:
Proposition 1.
Let$O$be an invex set and non-negative real-valued functionsuch that. Let$G : O → E C$be an I∙V∙F with$G ( x ) ≥ 0$, such that
for all$x ∈ O$. Then,$G$is-preinvex I∙V∙F on $O ,$if and only if,$G * ( x )$is-preinvex function and$G * ( x )$is-preincave function.
Proof.
The proof of this proposition is similar to the Theorem 3.7, see [73]. □
Proposition 2.
Let$O$be an invex set and non-negative real-valued functionsuch that. Let$G : O → E C +$be an I∙V∙F with$G ( x ) ≥ 0$, such that
for all$x ∈ O$. Then,$G$is-preincave I∙V∙F on $O ,$if and only if,$G * ( x )$and$G * ( x )$are-preincave and preinvex functions, respectively.
Proof.
The demonstration of the proof is analogous to Proposition 1. □

## 3. Results

In this section, using the Riemann integral operator, we achieve various modifications of the Hermite-Hadamard type inequality. We add a few quality and interesting remarks for the readers. The next step is to present a crucial 𝐻∙𝐻-inequality for -preinvex I∙V∙Fs via Riemann integrals.
Theorem 2.
Letbe a-preinvex I∙V∙F with non-negative real-valued functionsand$£ 1 ( 1 2 ) £ 2 ( 1 2 ) ≠ 0 ,$such thatfor all. If, then
Let be a -preincave I∙V∙F. Then, we have
Proof.
Let be a -preinvex I∙V∙F. Then, by hypothesis, we have
Therefore, we have
Then,
and it follows that
That is
Thus,
In a similar way as above, we have
Combining (16) and (17), we have
This completes the proof. □
Note that, if $G ( x )$ is -preincave I∙V∙F, then integral inequality (Equation (14)) is reversed.
Remark 2.
If$£ 2 ( o ) ≡ 1$, then Theorem 2 reduces to the result for$£ 1$-preinvex I∙V∙F, see[83]:
If $£ 1 ( o ) = o s$ and $£ 2 ( o ) ≡ 1$, then Theorem 2 reduces to the result for $s$-preinvex I∙V∙F, see [83]:
If $£ 1 ( o ) = o$ and $£ 2 ( o ) ≡ 1$, then Theorem 2 reduces to the result for preinvex I∙V∙F, see [84]:
If $£ 1 ( o ) = £ 2 ( o ) ≡ 1$, then Theorem 2 reduces to the result for $P$-I∙V∙F, see [83]:
If $G * ( x ) = G * ( x )$, then we from (14) obtain the classical integral inequality for $( £ 1 , £ 2 )$-preinvex functions.
If $G * ( x ) = G * ( x )$, $£ 2 ( o ) ≡ 1$, then from (14) we obtain the classical integral inequality for $( £ 1 , £ 2 )$-preinvex functions.
Note that, if $w ( q , ς ) = q − ς$, then the above integral inequalities reduce to classical ones.
Example 1.
We considerforand the I∙V∙Fdefined by. Since end point functions$G * ( x ) = 2 x 2 ,$$G * ( x ) = 4 x$are$( £ 1 , £ 2 )$-preinvex functions with respect to, then$G ( x )$is$( £ 1 , £ 2 )$-preinvex I∙V∙F with respect to. Now we compute the following
which means
$2 ≤ 8 3 ≤ 4 .$
Similarly, it can be easily show that
such that
that is
Hence, the Theorem 2 is verified.
Theorem 3.
(The second𝐻∙𝐻-Fejér inequality for. -preinvex I∙V∙Fs). Letbe a-preinvex I∙V∙F withand non-negative real-valued functions, such thatfor all. Ifandsymmetric with respect to, then
Proof.
Let $G$ be a -preinvex I∙V∙F. Then, we have
We also have
After adding (23) and (24), and integrating over we get
$∫ 0 1 G * ( ς + ( 1 − o ) w ( q , ς ) ) C ( ς + ( 1 − o ) w ( q , ς ) ) d o + ∫ 0 1 G * ( ς + o w ( q , ς ) ) C ( ς + o w ( q , ς ) ) d o ≤ ∫ 0 1 [ G * ( ς ) { £ 1 ( o ) £ 2 ( 1 − o ) C ( ς + ( 1 − o ) w ( q , ς ) ) + £ 1 ( 1 − o ) £ 2 ( o ) C ( ς + o w ( q , ς ) ) } + G * ( q ) { £ 1 ( 1 − o ) £ 2 ( o ) C ( ς + ( 1 − o ) w ( q , ς ) ) + £ 1 ( o ) £ 2 ( 1 − o ) C ( ς + o w ( q , ς ) ) } ] d o , ∫ 0 1 G * ( ς + o w ( q , ς ) ) C ( ς + o w ( q , ς ) ) d o + ∫ 0 1 G * ( ς + ( 1 − o ) w ( q , ς ) ) C ( ς + ( 1 − o ) w ( q , ς ) ) d o ≥ ∫ 0 1 [ G * ( ς ) { £ 1 ( o ) £ 2 ( 1 − o ) C ( ς + ( 1 − o ) w ( q , ς ) ) + £ 1 ( 1 − o ) £ 2 ( o ) C ( ς + o w ( q , ς ) ) } + G * ( q ) { £ 1 ( 1 − o ) £ 2 ( o ) C ( ς + ( 1 − o ) w ( q , ς ) ) + £ 1 ( o ) £ 2 ( 1 − o ) C ( ς + o w ( q , ς ) ) } ] d o .$
Since $C$ is symmetric, then
Since
From (25) and (26), we have
that is
hence
then we complete the proof. □
The following assumption is required to prove the next result regarding the bi-function $w : O × O → ℝ$ which is known as:
Condition C.
Let$O$be an invex set with respect to$w$. For anyand,
$w ( q , ς + o w ( q , ς ) ) = ( 1 − o ) w ( q , ς ) ,$
Clearly for $o$ = 0, we have $w ( q , ς )$ = 0 if and only if,, for all . For the applications of Condition C, see [79,80,81,82,83,84].
Theorem 4.
(The first𝐻∙𝐻- Fejér inequality for-preinvex I∙V∙Fs). Letbe a-preinvex I∙V∙F withand non-negative real-valued functions, such thatfor all. Ifandsymmetric with respect toand, and Condition C holds for$w$, the,
Proof.
Using Condition C, we can write
Since $G$ is a -preinvex then, we have
By multiplying (28) by $C ( ς + ( 1 − o ) w ( q , ς ) ) = C ( ς + o w ( q , ς ) )$ and integrating it by $o$ over we obtain
since
From (29) and (30), we have
from which we have
that is
And this completes the proof. □
Remark 3.
If$£ 2 ( o ) ≡ 1 ,$, then inequalities in Theorems 3 and 4 reduce for$£ 1$-preinvex I∙V∙Fs, see [83].
If $£ 1 ( o ) = o$ and $£ 2 ( o ) ≡ 1 ,$ , then inequalities in Theorems 3 and 4 reduce for preinvex I∙V∙Fs, see [84].
If in the Theorems 3 and 4 $£ 2 ( o ) ≡ 1$ and , then we obtain the appropriate theorems for $£ 1$-convex I∙V∙Fs, see [73].
If in the Theorems 3 and 4 $£ 1 ( o ) = o ,$ $£ 2 ( o ) ≡ 1$ and , then we obtain the appropriate theorems for convex I∙V∙Fs, see [73].
If $G * ( x ) = G * ( x )$ with $£ 2 ( o ) ≡ 1$, then Theorems 3 and 4 reduce to classical first and second 𝐻∙𝐻-Fejér inequalities for £-preinvex function, see [80].
If in the Theorems 3 and 4 $G * ( x ) = G * ( x )$ with $£ 2 ( o ) ≡ 1$ and , then we obtain the appropriate theorems for £-convex function.
If $C ( x ) = 1$, then combining Theorems 3 and 4, we get Theorem 2.
Example 2.
We consider$£ 2 ( o ) = 1$forand the I∙V∙Fdefined by,. Since end point functions$G * ( x ) ,$$G * ( x )$. are-preinvex functions$w ( y , x ) = y − x$, then$G ( x )$is-preinvex I∙V∙F. If
then we have
and
From (31) and (32), we have
Hence, Theorem 3 is verified.
For Theorem 4, we have
From (33) and (34), we have
Hence, Theorem 4 is verified.
Theorem 5.
Letbe two-preinvex I∙V∙Fs with non-negative real-valued functionssuch thatandfor all. If, then
where $ℳ ( ς , q ) = G ( ς ) × J ( ς ) + G ( q ) × J ( q ) ,$ $N ( ς , q ) = G ( ς ) × J ( q ) + G ( q ) × J ( ς )$ with and
Proof.
Since $G$ and $J$ both are -preinvex I∙V∙Fs on , then we have
We also have
From the definition of -preinvex I∙V∙Fs, it follows that $G ( x ) ≥ 0$ and $J ( x ) ≥ 0$, so
Integrating both sides of above inequality over [0, 1] we get
It follows that
that is
Thus,
and the theorem has been established. □
Theorem 6.
Letbe two-preinvex I∙V∙Fs with non-negative real valued functionsand$£ 1 ( 1 2 ) £ 2 ( 1 2 ) ≠ 0 ,$such thatandfor all. If, and Condition C hold for$w$, then
where $ℳ ( ς , q ) = G ( ς ) × J ( ς ) + G ( q ) × J ( q ) ,$ $N ( ς , q ) = G ( ς ) × J ( q ) + G ( q ) × J ( ς ) ,$ and and
Proof.
Using Condition C, we can write
By hypothesis, we have
integrating over we have
from which we have
that is
hence, the required result. □
Remark 4.
If$£ 2 ( o ) ≡ 1$,, then Theorems 5 and 6 reduce for$£ 1$-preinvex I∙V∙Fs, see [73].
If $£ 1 ( o ) = o$ and $£ 2 ( o ) ≡ 1 ,$ , then Theorems 5 and 6 reduce for preinvex I∙V∙Fs, see [73].
If in the above theorem $G * ( x ) = G * ( x )$, then we obtain the appropriate Theorems 5 and 6 for -preinvex functions, see [79].
If in Theorems 5 and 6 $G * ( x ) = G * ( x )$ with $£ 2 ( o ) ≡ 1 ,$ , then we obtain the appropriate theorems for $£ 1$-preinvex functions, see [80].
If in Theorems 5 and 6 $G * ( x ) = G * ( x )$, $w ( y , x ) = y − x ,$ $£ 1 ( o ) = o$ and $£ 2 ( o ) ≡ 1 ,$ , then we obtain the appropriate theorems for convex functions, see [81].
If in Theorems 5 and 6 $G * ( x ) = G * ( x )$, $w ( y , x ) = y − x ,$ $£ 1 ( o ) = o s$ and $£ 2 ( o ) ≡ 1 ,$ , , then we obtain the appropriate theorems for $s$-convex functions in the second sense, see [81].
Example 3.
We considerfor, and the I∙V∙Fsdefined byandSince end point functions$G * ( x ) = 2 x 2$and$G * ( x ) = 4 x$both are-preinvex functions, and$J * ( x ) = x$, and$J * ( x ) = 2 x$both are also-preinvex functions with respect to same, then$G$and$J$both are-preinvex I∙V∙Fs, respectively. Since$G * ( x ) = 2 x 2$and$G * ( x ) = 4 x$, and$J * ( x ) = x$, and$J * ( x ) = 2 x$, then
which means
Hence, Theorem 5 is verified.
For Theorem 6, we have
which means
Hence, Theorem 6 is demonstrated.

## 4. Conclusions

A useful method for introducing uncertainty into prediction processes is to use interval-valued functions. In order to establish the Hermite-Hadamard and Pachpatt-type inequalities, we first introduced a new idea of interval-valued harmonic convexity, i.e., a harmonically interval valued -preinvex function. Many of the definitions that already existed in the literature were generalized by our new concept. Thus, we contributed to the set-valued setting’s extension of several classical integral inequalities. To further explain the findings, some numerical examples were given.
This innovative approach could be applied to future presentations of various inequalities, such as those of the Hermite-Hadamard, Ostrowski, Hadamard-Mercer, Simpson, Fejér, and Bullen kinds. Numerous interval-valued LR convexities, fuzzy interval convexities, and CR convexities can be used to illustrate the aforementioned inequalities. Additionally, these results will be used to fractional calculus, coordinated interval-valued functions, quantum calculus, and other areas. Many mathematicians will be interested in examining how various types of interval-valued analyses may be applied to integral inequalities because these are the most active areas of research in the field of integral inequalities.

## Author Contributions

Conceptualization, M.B.K.; methodology, M.B.K.; validation, M.S.S.; formal analysis, M.S.S.; investigation, M.B.K.; resources, M.S.S.; data curation, J.E.M.-D.; writing—original draft preparation, M.B.K.; writing—review and editing, M.B.K. and M.S.S.; visualization, J.E.M.-D.; supervision, M.B.K. and M.A.N.; project administration, M.B.K. and M.A.N.; funding acquisition, M.S.S. and J.E.M.-D. All authors have read and agreed to the published version of the manuscript.

## Funding

This research received no external funding.

Not applicable.

## Acknowledgments

The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environments.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Moore, R.E. Interval Analysis; Prentice-Hall: Hoboken, NJ, USA, 1966. [Google Scholar]
2. Snyder, J. Interval analysis for computer graphics. SIGGRAPH Comput. Graph. 1992, 26, 121–130. [Google Scholar] [CrossRef]
3. Gasilov, N.A.; Emrah Amrahov, S. Solving a nonhomogeneous linear system of interval differential equations. Soft Comput. 2018, 22, 3817–3828. [Google Scholar] [CrossRef]
4. De Weerdt, E.; Chu, Q.P.; Mulder, J.A. Neural network output optimization using interval analysis. IEEE Trans. Neural Netw. 2009, 20, 638–653. [Google Scholar] [CrossRef] [PubMed][Green Version]
5. Rothwell, E.J.; Cloud, M.J. Automatic error analysis using intervals. IEEE Trans. Edu. 2011, 55, 9–15. [Google Scholar] [CrossRef]
6. Chalco-Cano, Y.; Rufián-Lizana, A.; Román-Flores, H.; Jiménez-Gamero, M.D. Calculus for interval-valued functions using generalized Hukuhara derivative and applications. Fuzzy Sets Syst. 2013, 219, 49–67. [Google Scholar] [CrossRef]
7. Chalco-Cano, Y.; Silva, G.N.; Rufián-Lizana, A. On the Newton method for solving fuzzy optimization problems. Fuzzy Sets Syst. 2015, 272, 60–69. [Google Scholar] [CrossRef]
8. Entani, T.; Inuiguchi, M. Pairwise comparison-based interval analysis for group decision aiding with multiple criteria. Fuzzy Sets Syst. 2015, 274, 79–96. [Google Scholar] [CrossRef]
9. Osuna-Gómez, R.; Chalco-Cano, Y.; Hernández-Jiménez, B.; Ruiz-Garzón, G. Optimality conditions for generalized differentiable interval-valued functions. Inf. Sci. 2015, 321, 136–146. [Google Scholar] [CrossRef]
10. Moore, R.E.; Kearfott, R.B.; Cloud, M.J. Introduction to Interval Analysis; Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, USA, 2009. [Google Scholar]
11. Khan, M.B.; Noor, M.A.; Macías-Díaz, J.E.; Soliman, M.S.; Zaini, H.G. Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation. Demonstr. Math. 2022, 55, 387–403. [Google Scholar] [CrossRef]
12. Khan, M.B.; Zaini, H.G.; Treanțǎ, S.; Soliman, M.S.; Nonlaopon, K. Riemann-Liouville Fractional Integral Inequalities for Generalized Pre-Invex Functions of Interval-Valued Settings Based upon Pseudo Order Relation. Mathematics 2022, 10, 204. [Google Scholar] [CrossRef]
13. Macías-Díaz, J.E.; Khan, M.B.; Noor, M.A.; Abd Allah, A.M.; Alghamdi, S.M. Hermite-Hadamard inequalities for generalized convex functions in interval-valued calculus. AIMS Math. 2022, 7, 4266–4292. [Google Scholar] [CrossRef]
14. Khan, M.B.; Noor, M.A.; Al-Bayatti, H.M.; Noor, K.I. Some New Inequalities for LR-Log-h-Convex Interval-Valued Functions by Means of Pseudo Order Relation. Appl. Math. 2021, 15, 459–470. [Google Scholar]
15. Khan, M.B.; Noor, M.A.; Abdeljawad, T.; Mousa, A.A.A.; Abdalla, B.; Alghamdi, S.M. LR-Preinvex Interval-Valued Functions and Riemann-Liouville Fractional Integral Inequalities. Fractal Fract. 2021, 5, 243. [Google Scholar] [CrossRef]
16. Khan, M.B.; Macías-Díaz, J.E.; Treanta, S.; Soliman, M.S.; Zaini, H.G. Hermite-Hadamard Inequalities in Fractional Calculus for Left and Right Harmonically Convex Functions via Interval-Valued Settings. Fractal Fract. 2022, 6, 178. [Google Scholar] [CrossRef]
17. Khan, M.B.; Noor, M.A.; Noor, K.I.; Nisar, K.S.; Ismail, K.A.; Elfasakhany, A. Some Inequalities for LR-(h1,h2)-Convex Interval-Valued Functions by Means of Pseudo Order Relation. Int. J. Comput. Intell. Syst. 2021, 14, 180. [Google Scholar] [CrossRef]
18. Chalco-Cano, Y.; Flores-Franuli’c, A.; Román-Flores, H. Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative. Comput. Appl. Math. 2012, 31, 457–472. [Google Scholar]
19. Costa, T.M.; Román-Flores, H. Some integral inequalities for fuzzy-interval-valued functions. Inf. Sci. 2017, 420, 110–125. [Google Scholar] [CrossRef]
20. Khan, M.B.; Cătaş, A.; Alsalami, O.M. Some New Estimates on Coordinates of Generalized Convex Interval-Valued Functions. Fractal Fract. 2022, 6, 415. [Google Scholar] [CrossRef]
21. Kalsoom, H.; Latif, M.A.; Khan, Z.A.; Vivas-Cortez, M. Some New Hermite-Hadamard-Fejér fractional type inequalities for 𝒽-convex and harmonically 𝒽-convex interval-valued Functions. Mathematics 2021, 10, 74. [Google Scholar] [CrossRef]
22. Liu, Z.-H.; Motreanu, D.; Zeng, S.-D. Generalized penalty and regularization method for differential variational- hemivariational inequalities. SIAM J. Optim. 2021, 31, 1158–1183. [Google Scholar] [CrossRef]
23. Liu, Y.-J.; Liu, Z.-H.; Wen, C.-F.; Yao, J.-C.; Zeng, S.-D. Existence of solutions for a class of noncoercive variational—Hemivariational inequalities arising in contact problems. Appl. Math. Optim. 2021, 84, 2037–2059. [Google Scholar] [CrossRef]
24. Zeng, S.-D.; Migorski, S.; Liu, Z.-H. Well-posedness, optimal control, and sensitivity analysis for a class of differential variational- hemivariational inequalities. SIAM J. Optim. 2021, 31, 2829–2862. [Google Scholar] [CrossRef]
25. Liu, Y.-J.; Liu, Z.-H.; Motreanu, D. Existence and approximated results of solutions for a class of nonlocal elliptic variational-hemivariational inequalities. Math. Methods Appl. Sci. 2020, 43, 9543–9556. [Google Scholar] [CrossRef]
26. Liu, Y.-J.; Liu, Z.-H.; Wen, C.-F. Existence of solutions for space-fractional parabolic hemivariational inequalities. Discret. Contin. Dyn. Syst. Ser. B 2019, 24, 1297–1307. [Google Scholar] [CrossRef][Green Version]
27. Liu, Z.-H.; Loi, N.V.; Obukhovskii, V. Existence and global bifurcation of periodic solutions to a class of differential variational inequalities. Internat. J. Bifur. Chaos Appl. Sci. Eng. 2013, 23, 1350125. [Google Scholar] [CrossRef]
28. Narges Hajiseyedazizi, S.; Samei, M.E.; Alzabut, J.; Chu, Y.-M. On multi-step methods for singular fractional q-integro-differential equations. Open Math. 2021, 19, 1378–1405. [Google Scholar] [CrossRef]
29. Jin, F.; Qian, Z.-S.; Chu, Y.-M.; ur Rahman, M. On nonlinear evolution model for drinking behavior under Caputo-Fabrizio derivative. J. Appl. Anal. Comput. 2022, 12, 790–806. [Google Scholar] [CrossRef]
30. Wang, F.-Z.; Khan, M.N.; Ahmad, I.; Ahmad, H.; Abu-Zinadah, H.; Chu, Y.-M. Numerical solution of traveling waves in chemical kinetics: Time-fractional fishers equations. Fractals 2022, 30, 2240051. [Google Scholar] [CrossRef]
31. Zhao, T.-H.; Bhayo, B.A.; Chu, Y.-M. Inequalities for generalized Grötzsch ring function. Comput. Methods Funct. Theory 2022, 22, 559–574. [Google Scholar] [CrossRef]
32. Iqbal, S.A.; Hafez, M.G.; Chu, Y.-M.; Park, C. Dynamical Analysis of nonautonomous RLC circuit with the absence and presence of Atangana-Baleanu fractional derivative. J. Appl. Anal. Comput. 2022, 12, 770–789. [Google Scholar] [CrossRef]
33. Huang, T.-R.; Chen, L.; Chu, Y.-M. Asymptotically sharp bounds for the complete p-elliptic integral of the first kind. Hokkaido Math. J. 2022, 51, 189–210. [Google Scholar] [CrossRef]
34. Zhao, T.-H.; Qian, W.-M.; Chu, Y.-M. On approximating the arc lemniscate functions. Indian J. Pure Appl. Math. 2022, 53, 316–329. [Google Scholar] [CrossRef]
35. Sana, G.; Khan, M.B.; Noor, M.A.; Mohammed, P.O.; Chu, Y.M. Harmonically convex fuzzy-interval-valued functions and fuzzy-interval Riemann-Liouville fractional integral inequalities. Int. J. Comput. Intell. Syst. 2021, 2021, 1809–1822. [Google Scholar] [CrossRef]
36. Khan, M.B.; Treanțǎ, S.; Soliman, M.S.; Nonlaopon, K.; Zaini, H.G. Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions. Fractal Fract. 2022, 6, 6. [Google Scholar] [CrossRef]
37. Khan, M.B.; Santos-García, G.; Noor, M.A.; Soliman, M.S. Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities. Chaos Solitons Fractals 2022, 164, 112692. [Google Scholar] [CrossRef]
38. Khan, M.B.; Noor, M.A.; Abdullah, L.; Chu, Y.M. Some new classes of preinvex fuzzy-interval-valued functions and inequalities. Int. J. Comput. Intell. Syst. 2021, 14, 1403–1418. [Google Scholar] [CrossRef]
39. Zhao, T.-H.; Zhou, B.-C.; Wang, M.-K.; Chu, Y.-M. On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, 2019, 42. [Google Scholar] [CrossRef][Green Version]
40. Zhao, T.-H.; Wang, M.-K.; Zhang, W.; Chu, Y.-M. Quadratic transformation inequalities for Gaussian hyper geometric function. J. Inequal. Appl. 2018, 2018, 251. [Google Scholar] [CrossRef][Green Version]
41. Chu, Y.-M.; Zhao, T.-H. Concavity of the error function with respect to Hölder means. Math. Inequal. Appl. 2016, 19, 589–595. [Google Scholar] [CrossRef]
42. Qian, W.-M.; Chu, H.-H.; Wang, M.-K.; Chu, Y.-M. Sharp inequalities for the Toader mean of order—1 in terms of other bivariate means. J. Math. Inequal. 2022, 16, 127–141. [Google Scholar] [CrossRef]
43. Zhao, T.-H.; Chu, H.-H.; Chu, Y.-M. Optimal Lehmer mean bounds for the nth power-type Toader mean of n = −1, 1, 3. J. Math. Inequal. 2022, 16, 157–168. [Google Scholar] [CrossRef]
44. Zhao, T.-H.; Wang, M.-K.; Dai, Y.-Q.; Chu, Y.-M. On the generalized power-type Toader mean. J. Math. Inequal. 2022, 16, 247–264. [Google Scholar] [CrossRef]
45. Zhao, T.-H.; Castillo, O.; Jahanshahi, H.; Yusuf, A.; Alassafi, M.O.; Alsaadi, F.E.; Chu, Y.-M. A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak. Appl. Comput. Math. 2021, 20, 160–176. [Google Scholar]
46. Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. On the bounds of the perimeter of an ellipse. Acta Math. Sci. 2022, 42B, 491–501. [Google Scholar] [CrossRef]
47. Zhao, T.-H.; Wang, M.-K.; Hai, G.-J.; Chu, Y.-M. Landen inequalities for Gaussian hypergeometric function. Rev. Real Acad. Cienc. Exactas Físicas Naturales. Ser. A Matemáticas RACSAM 2022, 116, 53. [Google Scholar] [CrossRef]
48. Wang, M.-K.; Hong, M.-Y.; Xu, Y.-F.; Shen, Z.-H.; Chu, Y.-M. Inequalities for generalized trigonometric and hyperbolic functions with one parameter. J. Math. Inequal. 2020, 14, 1–21. [Google Scholar] [CrossRef]
49. Zhao, T.-H.; Qian, W.-M.; Chu, Y.-M. Sharp power mean bounds for the tangent and hyperbolic sine means. J. Math. Inequal. 2021, 15, 1459–1472. [Google Scholar] [CrossRef]
50. Dragomir, S.S.; Pečarić, J.; Persson, L.E. Some inequalities of Hadamard type. Soochow J. Math. 1995, 21, 335–341. [Google Scholar]
51. Dragomir, S.S. Inequalities of Hermite-Hadamard type for functions of selfadjoint operators and matrices. J. Math. Inequal. 2017, 11, 241–259. [Google Scholar] [CrossRef][Green Version]
52. Latif, M. On Some New Inequalities of Hermite-Hadamard Type for Functions Whose Derivatives are s-convex in the Second Sense in the Absolute Value. Ukr. Math. J. 2016, 67, 1552–1571. [Google Scholar] [CrossRef]
53. Noor, M.A.; Cristescu, G.; Awan, M.U. Generalized Fractional Hermite-Hadamard Inequalities for Twice Differentiable s-convex Functions. Filomat 2015, 29, 807–815. [Google Scholar] [CrossRef]
54. Noor, M.A.; Noor, K.I.; Awan, M.U.; Li, J. On Hermite-Hadamard Inequalities for 𝒽-Preinvex Functions. Filomat 2014, 28, 1463–1474. [Google Scholar] [CrossRef]
55. Liu, P.; Khan, M.B.; Noor, M.A.; Noor, K.I. New Hermite-Hadamard and Jensen inequalities for log-s-convex fuzzy-interval-valued functions in the second sense. Complex. Intell. Syst. 2021, 8, 413–427. [Google Scholar] [CrossRef]
56. Khan, M.B.; Treanțǎ, S.; Budak, H. Generalized p-Convex Fuzzy-Interval-Valued Functions and Inequalities Based upon the Fuzzy-Order Relation. Fractal Fract. 2022, 6, 63. [Google Scholar] [CrossRef]
57. Santos-García, G.; Khan, M.B.; Alrweili, H.; Alahmadi, A.A.; Ghoneim, S.S. Hermite-Hadamard and Pachpatte type inequalities for coordinated preinvex fuzzy-interval-valued functions pertaining to a fuzzy-interval double integral operator. Mathematics 2022, 10, 2756. [Google Scholar] [CrossRef]
58. Macías-Díaz, J.E.; Khan, M.B.; Alrweili, H.; Soliman, M.S. Some Fuzzy Inequalities for Harmonically s-Convex Fuzzy Number Valued Functions in the Second Sense Integral. Symmetry 2022, 14, 1639. [Google Scholar] [CrossRef]
59. Zhao, T.-H.; He, Z.-Y.; Chu, Y.-M. Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals. Comput. Methods Funct. Theory 2021, 21, 413–426. [Google Scholar] [CrossRef]
60. Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. Concavity and bounds involving generalized elliptic integral of the first kind. J. Math. Inequal. 2021, 15, 701–724. [Google Scholar] [CrossRef]
61. Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. Monotonicity and convexity involving generalized elliptic integral of the first kind. Rev. De La Real Acad. De Cienc. Exactas Físicas Y Naturales. Ser. A Matemáticas RACSAM 2021, 115, 46. [Google Scholar] [CrossRef]
62. Chu, H.-H.; Zhao, T.-H.; Chu, Y.-M. Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contra harmonic means. Math. Slovaca 2020, 70, 1097–1112. [Google Scholar] [CrossRef]
63. İşcan, İ. Hermite-Hadamard type inequalities for harmonically convex functions. Hacet. J. Math. Stat. 2014, 43, 935–942. [Google Scholar] [CrossRef]
64. Noor, M.A.; Noor, K.I.; Awan, M.U.; Costache, S. Some integral inequalities for harmonically 𝒽-convex functions. Politehn. Univ. Buchar. Sci. Bull. Ser. A Appl. Math. Phys. 2015, 77, 5–16. [Google Scholar]
65. Chalco-Cano, Y.; Lodwick, W.A. Condori-Equice. Ostrowski type inequalities and applications in numerical integration for interval-valued functions. Soft Comput. 2015, 19, 3293–3300. [Google Scholar] [CrossRef]
66. Román-Flores, H.; Chalco-Cano, Y.; Lodwick, W.A. Some integral inequalities for interval-valued functions. Comput. Appl. Math. 2018, 37, 1306–1318. [Google Scholar] [CrossRef]
67. Costa, T.M.; Román-Flores, H.; Chalco-Cano, Y. Opial-type inequalities for interval-valued functions. Fuzzy Set. Syst. 2019, 358, 48–63. [Google Scholar] [CrossRef]
68. Zhao, D.; Ali, M.A.; Murtaza, G.; Zhang, Z. On the Hermite-Hadamard inequalities for interval-valued coordinated convex functions. Adv. Differ. Equ. 2020, 2020, 570. [Google Scholar] [CrossRef]
69. Nwaeze, E.R.; Khan, M.A.; Chu, Y.M. Fractional inclusions of the Hermite-Hadamard type for m-polynomial convex intervalvalued functions. Adv. Differ. Equ. 2020, 2020, 507. [Google Scholar] [CrossRef]
70. Sharma, N.; Singh, S.K.; Mishra, S.K.; Hamdi, A. Hermite-Hadamard type inequalities for interval-valued preinvex functions via Riemann-Liouville fractional integrals. J. Inequal. Appl. 2021, 2021, 98. [Google Scholar] [CrossRef]
71. Saeed, T.; Khan, M.B.; Treanțǎ, S.; Alsulami, H.H.; Alhodaly, M.S. Interval Fejér-Type Inequalities for Left and Right-λ-Preinvex Functions in Interval-Valued Settings. Axioms 2022, 11, 368. [Google Scholar] [CrossRef]
72. Lai, K.K.; Bisht, J.; Sharma, N.; Mishra, S.K. Hermite-Hadamard-Type Fractional Inclusions for Interval-Valued Preinvex Functions. Mathematics 2022, 10, 264. [Google Scholar] [CrossRef]
73. Zhao, D.; An, T.; Ye, G.; Liu, W. New Jensen and Hermite-Hadamard type inequalities for 𝒽-convex interval-valued functions. J. Inequal. Appl. 2018, 302, 302. [Google Scholar] [CrossRef][Green Version]
74. Zhao, D.; An, T.; Ye, G.; Torres, D.F. On Hermite-Hadamard type inequalities for harmonical 𝒽-convex interval-valued functions. arXiv 2019, arXiv:1911.06900. [Google Scholar]
75. An, Y.; Ye, G.; Zhao, D.; Liu, W. Hermite-Hadamard type inequalities for interval (𝒽1, 𝒽2)-convex functions. Mathematics 2019, 7, 436. [Google Scholar] [CrossRef][Green Version]
76. Liu, R.; Xu, R. Hermite-Hadamard type inequalities for harmonical (𝒽1, 𝒽2) convex interval-valued functions. Math. Found. Comput. 2021, 4, 89. [Google Scholar] [CrossRef]
77. Almutairi, O.; Kiliçman, A.A.A. Some integral inequalities for 𝒽-Godunova-Levin preinvexity. Symmetry 2019, 11, 1500. [Google Scholar] [CrossRef][Green Version]
78. Bai, H.; Saleem, M.S.; Nazeer, W.; Zahoor, M.S.; Zhao, T. Hermite-Hadamard and Jensen type inequalities for interval nonconvex function. J. Math. 2020, 2020, 3945384. [Google Scholar] [CrossRef]
79. Costa, T. Jensen’s inequality type integral for fuzzy-interval-valued functions. Fuzzy Sets Syst. 2017, 327, 31–47. [Google Scholar] [CrossRef]
80. Noor, M.A.; Noor, K.I.; Rashid, S. Some new classes of preinvex functions and inequalities. Mathematics 2019, 7, 29. [Google Scholar] [CrossRef][Green Version]
81. Matłoka, M. Inequalities for h-preinvex functions. Appl. Math. Comput. 2014, 234, 52–57. [Google Scholar] [CrossRef]
82. Kirmaci, U.S. Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 2004, 147, 137–146. [Google Scholar] [CrossRef]
83. Kirmaci, U.S.; Bakula, M.K.; Ozdemir, M.E.; Pečarić, J. Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 2007, 193, 26–35. [Google Scholar] [CrossRef]
84. Khan, M.B.; Treanțǎ, S.; Soliman, M.S. Generalized Preinvex Interval-Valued Functions and Related Hermite-Hadamard Type Inequalities. Symmetry 2022, 14, 1901. [Google Scholar] [CrossRef]
85. Zhao, T.-H.; He, Z.-Y.; Chu, Y.-M. On some refinements for inequalities involving zero-balanced hyper geometric function. AIMS Math. 2020, 5, 6479–6495. [Google Scholar] [CrossRef]
86. Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. A sharp double inequality involving generalized complete elliptic integral of the first kind. AIMS Math. 2020, 5, 4512–4528. [Google Scholar] [CrossRef]
87. Zhao, T.-H.; Shi, L.; Chu, Y.-M. Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means. Rev. Real Acad. Cienc. Exactas Físicas Naturales. Ser. A Matemáticas RACSAM 2020, 114, 96. [Google Scholar] [CrossRef]
88. Khan, M.B.; Noor, M.A.; Zaini, H.G.; Santos-García, G.; Soliman, M.S. The New Versions of Hermite-Hadamard Inequalities for Pre-invex Fuzzy-Interval-Valued Mappings via Fuzzy Riemann Integrals. Int. J. Comput. Intell. Syst. 2022, 15, 66. [Google Scholar] [CrossRef]
89. Khan, M.B.; Noor, M.A.; Al-Shomrani, M.M.; Abdullah, L. Some Novel Inequalities for LR-h-Convex Interval-Valued Functions by Means of Pseudo Order Relation. Math. Meth. Appl. Sci. 2022, 45, 1310–1340. [Google Scholar] [CrossRef]
90. Khan, M.B.; Mohammed, P.O.; Noor, M.A.; Alsharif, A.M.; Noor, K.I. New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation. AIMS Math. 2021, 6, 10964–10988. [Google Scholar] [CrossRef]
91. Zeng, S.-D.; Migórski, S.; Liu, Z.-H. Nonstationary incompressible Navier-Stokes system governed by a quasilinear reaction-diffusion equation. Sci. Sin. Math. 2022, 52, 331–354. [Google Scholar]
92. Liu, Z.-H.; Sofonea, M.T. Differential quasivariational inequalities in contact mechanics. Math. Mech. Solids. 2019, 24, 845–861. [Google Scholar] [CrossRef]
93. Zeng, S.-D.; Migórski, S.; Liu, Z.-H.; Yao, J.-C. Convergence of a generalized penalty method for variational-hemivariational inequalities. Commun. Nonlinear Sci. Numer. Simul. 2021, 92, 105476. [Google Scholar] [CrossRef]
94. Li, X.-W.; Li, Y.-X.; Liu, Z.-H.; Li, J. Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions. Fract. Calc. Appl. Anal. 2018, 21, 1439–1470. [Google Scholar] [CrossRef]
95. Liu, Z.-H.; Papageorgiou, N.S. Positive solutions for resonant (p,q)-equations with convection. Adv. Nonlinear Anal. 2021, 10, 217–232. [Google Scholar] [CrossRef]
96. Liu, Y.-J.; Liu, Z.-H.; Motreanu, D. Differential inclusion problems with convolution and discontinuous nonlinearities. Evol. Equ. Control Theory 2020, 9, 1057–1071. [Google Scholar] [CrossRef]
97. Liu, Z.-H.; Papageorgiou, N.S. Double phase Dirichlet problems with unilateral constraints. J. Differ. Equ. 2022, 316, 249–269. [Google Scholar] [CrossRef]
98. Liu, Z.-H.; Papageorgiou, N.S. Anisotropic (p,q)-equations with competition phenomena. Acta Math. Sci. 2022, 42B, 299–322. [Google Scholar] [CrossRef]
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Khan, M.B.; Macías-Díaz, J.E.; Soliman, M.S.; Noor, M.A. Some New Integral Inequalities for Generalized Preinvex Functions in Interval-Valued Settings. Axioms 2022, 11, 622. https://doi.org/10.3390/axioms11110622

AMA Style

Khan MB, Macías-Díaz JE, Soliman MS, Noor MA. Some New Integral Inequalities for Generalized Preinvex Functions in Interval-Valued Settings. Axioms. 2022; 11(11):622. https://doi.org/10.3390/axioms11110622

Chicago/Turabian Style

Khan, Muhammad Bilal, Jorge E. Macías-Díaz, Mohamed S. Soliman, and Muhammad Aslam Noor. 2022. "Some New Integral Inequalities for Generalized Preinvex Functions in Interval-Valued Settings" Axioms 11, no. 11: 622. https://doi.org/10.3390/axioms11110622

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