Some New Fractional Inequalities Involving Convex Functions and Generalized Fractional Integral Operator

Neamah, Majid K.; Ibrahim, Alawiah; Mehdy, Hala Shaker; Redhwan, Saleh S.; Abdo, Mohammed S.

doi:https://doi.org/10.1155/2022/2350193

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 2350193 | https://doi.org/10.1155/2022/2350193

Some New Fractional Inequalities Involving Convex Functions and Generalized Fractional Integral Operator

Majid K. Neamah,^1,2Alawiah Ibrahim,²Hala Shaker Mehdy,³Saleh S. Redhwan,⁴and Mohammed S. Abdo⁵

Academic Editor: Humberto Rafeiro

Received20 Jan 2022

Accepted28 Feb 2022

Published21 Mar 2022

Abstract

In this manuscript, we are getting some novel inequalities for convex functions by a new generalized fractional integral operator setting. Our results are established by merging the -Riemann-Liouville fractional integral operator with the generalized Katugampola fractional integral operator. Certain special instances of our main results are considered. The detailed results extend and generalize some of the present results by applying some special values to the parameters.

1. Introduction

Chebyshev [1] presented the celebrated functional described by where the functions and are integrable on If and are synchronous, i.e., for then,.

The relation (1) has stood out for some researchers because of the different applications in numerical quadrature, statistical problems, probability, and transform theory. Among those applications, the relation (1) was utilized to yield various integral inequalities (see, e.g., [2–6], for extremely late work, see additionally [7–10]).

There is one more appealing and useful inequality, namely, the Pólya-Szegö inequality [11], which establishes the essential key of motivation in our study, which we can indicate as follows where and for some and for each

In [12], Dragomir and Diamond introduced the following Grüss type integral inequality by the Pólya-Szegö result, here, and for

The next integral inequalities have been proved by Ngo [13], where and are a positive continuous on such that

In this regard, Liu et al. [14] introduced the subsequent inequality

with where and are a positive continuous on and for

Since one of the primary inspiration points of fractional analysis is getting more general and valuable fundamental integral operators, the generalized fractional integral operator is a decent tool to sum up numerous past investigations and results, see [15–18]. Essentially, in inequality theory, scientists utilize such broad operators to generalize and extend their inequalities.

Lately, these fractional integral operators have been considered and used to broaden particularly Grüss, Chebychev-Grüss, Pólya-Szegö, Gronwall, Minkowski, and Hermite-Hadamard type inequalities. For additional subtleties, Agarwal [19] proved some fractional integral inequalities with Hadamard’s fractional integral operators. Ntouyas et al. [20] established certain Chebyshev type inequalities involving Hadamard’s fractional integral operators. Some Grüss type inequalities under -Riemann-Liouville fractional integral operators have been investigated by Set et al. [21]. In [22], the authors introduced some Pólya-Szegö type inequalities by Hadamard -fractional integral operators. A new version for the Gronwall type inequality involving generalized proportional fractional integral operators was presented by Alzabut et al. [23]. Rahman et al. [24] established reverse Minkowski inequalities with generalized proportional fractional integral operators. Many Chebyshev type inequalities with generalized conformable fractional integral operators have been discussed by Nisar et al. [25].

In this regard, Dahmani [26] established some new inequalities for convex functions involving Riemann-Liouville fractional integral operator. Jleli et al. [27] obtained new Hermite-Hadamard type inequalities for convex functions via generalized fractional integral operators. Some Hermite-Hadamard type inequalities for -Riemann-Liouville fractional integral operators were obtained by Agarwal et al. [28]. The authors in [29] established certain Polya-Szego type inequalities involving generalized Katugampola fractional integral operator.

Motivated by the above works and discussions, in this manuscript, we establish certain novel inequalities for convex functions under a new generalized fractional integral operator which integrates the two proposed fractional integral operators in [30, 31]. Moreover, we consider certain special cases of our main results. The results obtained extend and generalize some of the existing results by substituting some parameters.

The manuscript is marshaled as follows: Section 2 presents some main definitions and results. The acquired results are presented in Section 3. The last section concludes the manuscript.

2. Preliminaries

Let us first present the essential definitions and properties of fractional analysis that will be frequently used in this study.

Definition 1 (see [15]). The right and left-sided Riemann-Liouville fractional integral operators of order of are defined by

Definition 2 (see [30]). For , and the left-sided -Riemann-Liouville fractional integral operator is defined as

Definition 3 (see [31]). For and the left-sided generalized Katugampola fractional integral operator is defined as The subsequent important results have been established by Liu et al. [32].

Theorem 4 (see [32]). Let and be two positive continuous functions with on . Assume that the functions and are decreasing and increasing, respectively. If the function is a convex with , then

Theorem 5 (see [32]). Let , , and be positive continuous functions with on Let also are increasing functions and is a decreasing function. If the function is a convex with , then

3. Main Results

In this section, we will establish some new tolerances for the convex functions under a new fractional integral operator which combine together the two operators proposed in [30, 31].

Definition 6. Let , and Then, the generalized fractional integral operator of order for a continuous function is defined as

Remark 7. (1)Setting in Eq. (16), the fractional integral operator Eq. (16) reduces to the generalized Katugampola fractional integral operator defined by Eq. (13)(2)Setting , and in Eq. (16), the fractional integral operator Eq. (16) reduces to the generalized fractional integral operator defined by Eq. (12)(3)Setting , and in Eq. (16), with L’Hôpital’s rule, the fractional integral operator Eq. (16) reduces to the Hadamard fractional integral operator, namely,(4)Setting and in Eq. (16), the fractional integral operator Eq. (16) reduces to the Erdélyi-Kober fractional integral operator (see [15])(5)Setting , and in Eq. (16), the fractional integral operator Eq. (16) reduces to the -Riemann-Liouville fractional integral operator, i.e.,(6)Setting , and in Eq. (16), the fractional integral operator Eq. (16) reduces to the Riemann-Liouville fractional integral operator defined byNow, we are ready to provide the inequalities for convex functions by using the considered fractional integral operator defined by Eq. (16).

Theorem 8. Let and be positive continuous functions with on . If and are increasing and decreasing on , respectively, then for any convex function with we have where are as in Definition 6.

Proof. By the hypotheses of theorem, is convex with . Then the function is increasing. Since is an increasing function, thus, is an increasing function too.
Obviously, is a decreasing function. Therefore, for each , we have It follows that Multiplying Eq. (22) by , we obtain Multiplying Eq. (23) by and integrating Eq. (23) with respect to over we get Hence Again, multiplying Eq. (25) by and integrating Eq. (25) with respect to over we obtain Consequently, we have Since for all and the function is an increasing, thus, for we have Multiplying both sides of Eq. (28) by then integrating with respect to over , we get As per Eq. (16) can be written Eq. (29) as follows Hence, from Eq. (27) and Eq. (30), we obtain the desired result Eq. (20).

Remark 9. (i)When , and in Theorem 8, we get the result (Theorem 3.1) proved by Dahmani [26].(ii)When , and in Theorem 8, we recapture Theorem 4(iii)In Theorem 8, if we replace the operator with the generalized proportional fractional integral operator, then, we obtain the result (Theorem 3.1) proved by Neamah and Ibrahim [33].

Theorem 10. Let and be positive continuous functions with on . If is increasing and is decreasing on , then for any convex function with we have where and are as in Definition 6.

Proof. Thanks to the hypotheses of theorem, is convex with . Thus, is an increasing function. Furthermore, since is increasing function, the function is increasing. Distinctly, is decreasing function. Thus, by multiplying Eq. (25) by and integrating the nascent identity with respect to over , we obtain Consequently, the inequalities Eq. (30) and Eq. (32) give the inequality Eq. (31).

Remark 11. (i)Applying Theorem 10 for , we get Theorem 8(ii)Applying Theorem 10 for , and , we get the result (Theorem 8) proved by Dahmani [26].(iii)Applying Theorem 10 for , and , we get Theorem 4

Theorem 12. Let , , and be three positive continuous functions and on If is decreasing, and are increasing, and is convex function with . Then

Proof. In view of conditions of theorem, is convex with . Thus, is increasing. Besides, from the increasing of is increasing. Obviously, the function is decreasing. So, for all and we have Then, Multiplying Eq. (35) by then integrating the resulting inequality with respect to over we get Hence, With the same arguments as above for Eq. (37), we obtain It follows that Further, since on then using fact that the function is increasing, we can write By some previously repeated procedure, the inequality Eq. (40) leads to Given Eq. (16), the inequality Eq. (41) can be written as Therefore, from Eq. (42) and Eq. (39), we get Eq. (33), which completes the proof.

Remark 13. (1)Applying Theorem 12 for and , we get the result (Theorem 10) proved by Dahmani [26](2)It is noteworthy that Theorem 5 is a special case of Theorem 12 when , and

Theorem 14. Let , , and be three positive continuous functions and on . If is decreasing, and are increasing on , and is convex function such that Then, we have

Proof. Applying on both sides of Eq. (37), then integrating the resulting inequality with respect to over , we get Since on then using fact that the function is increasing we have Multiplying Eq. (45) by then integration with respect to over we obtain Following similar arguments as mentioned earlier, we conclude that Hence, by virtue of Eq. (44), Eq. (47), and Eq. (48), we obtain Eq. (43). Thus, the proof is completed.

Remark 15. (i)Applying Theorem 14 for , we obtain Theorem 12(ii)Applying Theorem 14 for and , we obtain Theorem 12 proved by Dahmani [26]

4. Conclusions

In this work, we have established certain Pólya-Szegö inequalities by using convex functions under a new generalized fractional integral operator. More precisely, some new results have been established by merging the -Riemann-Liouville fractional integral operator with the generalized Katugampola fractional integral operator. Moreover, we have introduced several new special results that cover many classical fractional integral operators.

In future work, it will be very interesting to study the inequalities considered in this work under a more general fractional integral operator in terms of another function , precisely, we hint to , and this is what we will think about in the next work.

Data Availability

Data are available upon request.

Conflicts of Interest

No conflicts of interest are related to this work.

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Copyright © 2022 Majid K. Neamah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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