Fractional Integral Inequalities concerning Extended Bessel Function in the Kernel

Abstract

The major purpose of this paper is to use the fractional integral operator in terms of extended generalized Bessel function to estimate new fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions. We prove a set of inequalities for the fractional integral operator in terms of extended generalized Bessel function integrals with one and two parameters. Also, we discussed some special cases of the obtained result.

1. Introduction

Integral inequalities are particularly essential in the subject of fractional differential equations. In the last few decades, this field has gained more recognition. The important developments in this field have been investigated by various researchers (for example, [1–3]). In [4, 5], the authors presented Grüss-type inequalities and some new inequalities by utilizing Riemann–Liouville (RL) fractional integrals. In [6], certain inequalities for the generalized -fractional integral operator were proposed. The generalized Hermite–Hadamard-type inequalities via fractional integral operators were proposed in [7]. By using a family of positive functions, Dahmani [8] developed several fractional integral inequalities. For the application of fractional integral inequalities, we suggest [9–15]. In recent decades, researchers have devoted their time and effort to the topic of fractional calculus due to its wide applications in a variety of sciences. In [16–21], some numerous developments of fractional integral operators and their applications in various domains can be found. As a result of these novel fractional integral operators, future research will focus on developing new ideas for connecting fractional operators by addressing new fractional integral inequalities. The interested readers are referred to [22–24]. Recently, Kashuri and Liko [25] and Luo et al. [26] presented certain remarkable integral inequalities for the generalized fractional integrals.

Ngo et al. [27] proposed the following inequalities:where and the function is positive and continuous on with

Liu et al. [28] proposed the following inequalities:where , and the function is positive and continuous on with

Liu et al. [29] presented the following two theorems for integral inequalities.

Theorem 1. Let and be two positive and continuous functions on withon, such that is decreasing and is increasing. Assume that is convex function with. Then, the following inequality holds:

Theorem 2. Let, , andbe positive and continuous functions on,, withon, such that is decreasing and and are increasing. Assume thatis convex function with. Then, the following inequality holds:

The inequalities in (1), (2), and (4) and their several extensions have been widely studied by the researchers [30, 31]. Furthermore, the research of fractional integral inequality is also of extreme significance.

Tilahun et al. [32] proposed an extended form of the fractional integral containing Bessel function in the kernel bywhere is the extended Bessel function, defined bywhere

Remark 1. If we consider in (8), then we get

Remark 2. If we consider , then (11) reduces to the following operator defined byIn this paper, we present fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions by utilizing integral operator (8).

2. Main Results

In this section, we present integral inequalities by employing the fractional integral concerning the extended Bessel function in the kernel.

Theorem 3. Let and be two continuous and positive functions on and on . Assume that , , is decreasing and is an increasing function on . Then, for any convex function with , the following integral inequality holds:where with,,, and .

Proof. By the hypothesis of the theorem, since the function is convex such that , the function is increasing. Also, is increasing; therefore, is increasing. Obviously, the function is decreasing. Therefore, for all , we haveIt can also be written asThe multiplication of (15) by yieldsNow, first conducting the multiplication of (16) by and then integrating the estimated inequality with respect to over , , we getIn view of (8), we getAgain, conducting the multiplication of (18) by and then integrating the resulting inequality with respect to over , , we getIn view of (8), it can be written asSince by hypothesis, on and is an increasing function, for , , we haveThus, multiplying (21) by and then integrating the estimated inequality with respect to over , , yieldswhich in view of equation (8) follows thatBy combining (20) and (23), we get (13).
By considering in Theorem 3, we get the following result for the operator given in (11).

Corollary 1. Letandbe two continuous and positive functions onand on . Assume that , , is decreasing and is an increasing function on. Then, for any convex function with , the following integral inequality holds:where with and .

Remark 3. By considering in Theorem 3, we get the result of Dahmani [33].

Remark 4. Applying Theorem 3 for , , and , we get Theorem 1.

Theorem 4. Letandbe continuous and positive on, , andon. Assume that,, is decreasing function andis increasing function on. Then, for any convex function with , the following inequality holds:where with,,,, and .

Proof. By the hypothesis of the theorem, is convex function and , and the function is increasing. Also, since is increasing, is obviously increasing function. As is decreasing function, for all , . Thus, by taking product of (18) and and integrating the resulting inequality with respect to over , , we getAlso, on the other hand, the function on and is an increasing function. Therefore, for, , , we haveThe multiplication of (27) by and integration of the resultant inequality with respect to over , , yieldswhich with the aid of (8) follows thatSimilarly, one can getBy combining (26), (29), and (30), we get the required inequality.
By considering in Theorem 4, we get the following result for the operator given in (11).

Corollary 2. Letandbe continuous and positive on,, and on . Assume that,, is decreasing function and is increasing function on. Then, for any convex function with , the following inequality holds:where with,, and.

Remark 5. By considering in Theorem 4, we get the result of Dahmani [33].

Remark 6. Applying Theorem 4 for , , and , we get Theorem 1.

Remark 7. Applying Theorem 4 for , we get Theorem 3.

Theorem 5. Let ,, andbe continuous and positive functions on,, andon. If the function,, is decreasing,andare increasing functions on. Then, ifis any convex function with, the following inequality holds:where , with ,,, and .

Proof. By the hypothesis of the theorem, is convex function such that and the function is increasing. Also, since is increasing, is increasing. It is clear that is decreasing. Thus, for all , , we haveIt can be written asThe multiplication of (34) by and integration of the resultant inequality with respect to over , , yieldsWith the aid of (8), we can write the above equation asAgain, the multiplication of (36) by and integration of the estimated inequality with respect to over , , and then applying (8) yieldsIt can also be written asHence, by using (23) and (38), we get the required assertion.
By considering in Theorem 5, we get the following result for the operator given in (11).

Corollary 3. Let,, andbe continuous and positive functions on,, and on . If the function,, is decreasing, and are increasing functions on. Then, if is any convex function with, the following inequality holds:where , with and .

Remark 8. By considering in Theorem 5, we get the result of Dahmani [33].

Remark 9. Applying Theorem 5 for , and , we get Theorem 2.

Theorem 6. Let ,, and be continuous and positive functions on,, andon . Ifis decreasing, and are increasing functions on. Let be any convex function with . Then, the following inequality holds:where with,,,, and.

Proof. Taking multiplication of (36) by and integrating the inequality with respect to over , , we obtainHence, by using (29), (30), and (41), we get the desired assertion.
By considering in Theorem 6, we get the following result for the operator given in (11).

Corollary 4. Let,, and be continuous and positive functions on,, and on . Ifis decreasing, and are increasing functions on. Let be any convex function with . Then, the following inequality holds:where with ,, and .

Remark 10. By considering in Theorem 6, we get the result of Dahmani [33].

Remark 11. Applying Theorem 6 for , , and , we get Theorem 2.

Remark 12. Applying Theorem 6 for , we get Theorem 5.

3. Concluding Remarks

In this present investigation, we have investigated some new generalized integral inequalities for the operator involving generalized extended Bessel function in the kernel. The integral inequalities presented in this paper are more general than the existing inequalities given in the literature. The special cases of the obtained result can be found in the work of [29, 33]. One can obtain various types of integral inequalities by employing integral operator (8).

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally, and they read and approved the final manuscript for publication.