Some New Inequalities for p-Convex Functions via a K-Fractional Conformable Integral

Dou, Yan; Saleem, Muhammad Shoaib; Anwar, Nimra; Gao, Haiping

doi:https://doi.org/10.1155/2022/5406897

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5406897 | https://doi.org/10.1155/2022/5406897

Some New Inequalities for p-Convex Functions via a K-Fractional Conformable Integral

Yan Dou,¹Muhammad Shoaib Saleem,²Nimra Anwar,²and Haiping Gao³

Academic Editor: Xiaolong Qin

Received05 Apr 2022

Revised23 May 2022

Accepted01 Jun 2022

Published18 Jul 2022

Abstract

The intention of this paper is to develop some new Hermite–Jensen–Mercer type inequalities for convex functions via fractional conformable integrals. Several existing results are also discussed which can be deduced from our results.

1. Introduction

In the last few decades, the theory of convexity got a remarkable development due to its wide range of applications in optimization problems and other applied sciences [1, 2]. It has several important and interesting properties [3]. Because of its importance, several generalization and extensions have been established by a number of researchers [4]. The definition of a classical convex function is as given below:

A function : I is said to be convex ifholds for all x, y and .

Furthermore, the theory of convexity is interlinked to the theory of inequalities due to the nature of its definition having geometric interpretation. Several classical inequalities have been derived by many authors via convex functions, for example, see [5, 6]. One of them is Jensen’s inequality, see [7], which can be stated as follows.

Definition 1 (See [26]). Let 0 and be a nonnegative weight with and be a convex function on the interval . Then, the inequalityholds for all and .

Another important inequality for convex functions is the Hermite–Hadamard inequality [8] proposed by Hermite and Hadamard in 1883, which was published in 1893, and is considered as one of the fundamental tools in mathematical literature due to its vast applications.

The Hermite–Hadamard inequality states that if is a convex function defined on and such that , then

It is also seldom called Hadamard’s inequality. A function is said to be convex if and only if it satisfies the Hermite–Hadamard inequality. Many new inequalities for convex functions have been established with the help of the Hermite–Hadamard inequality. These inequalities played an important role in both pure and applied mathematics. There are many recent studies which deal with several extensions, generalizations, and modifications of the Hermite–Hadamard inequality, for example, see [9, 10].

In 2003, Mercer [11] proved the following variant of Jensen’s inequality.

Theorem 1. If is a convex function in , thenholds for all and with .

It is known as the Jensen–Mercer inequality in literature. Recently, notable contributions have been made to the Jensen–Mercer inequality [12, 13].

Fractional calculus is a branch of mathematical analysis which deals with the derivatives and integrals for a non-integer order and is as old as classical calculus. In the last forty years or so, it has become a rapidly growing area, mainly due to its extensively significant applications in widespread fields of engineering, biological, and physical sciences. Thousands of research papers have been published in this regard, for example, see [14, 15] and the references therein.

Before we proceed further, let us have a brief review of previously known concepts which will be helpful in proving the results of this paper.

2. Preliminaries

A generalization of the convex function which is frequently used in convex analysis is the , convexity. In Reference [16], Zhang and Wan introduced the concept of , convexity as follows.

Definition 2. Let be a real interval and ; then, is said to be a , convex set, ifholds for all , and , where or , , , and .

Definition 3. Let be a real interval and . A function is said to be a , convex function ifholds for all , and .

If the above inequality is reversed, then is said to be concave. It can also be easily seen that for , the , convexity reduces to ordinary convexity of functions.

In References [17, 18], the Riemann–Liouville fractional integrals are defined as follows:

Definition 4. Let be a finite interval on the real axis and . The right- and left-sided Reimann–Liouville fractional integrals and of order with are defined by the following:andrespectively.

Now, we define the Hadamard fractional integrals, which were introduced in [19].

Definition 5. Let , ; then, the left- and right-sided Hadamard fractional integrals of order are defined as follows:andrespectively.

In [20], Katugampola introduced new fractional integral operators as follows.

Definition 6. Let be a finite interval. Then, the left- and right-sided Katugampola fractional integrals of order of are defined by the following:andwith and .

In [21], Jarad introduced the new fractional conformable integrals operators as follows,

Definition 7. Let , . Then, the left and right sided fractional conformable integral operators are defined by the following:andrespectively.

Remark 1. (1)The fractional integral (13) coincides with (7), (9), and (11), if we choose , , , , and , respectively.(2)The fractional integral (14) coincides with (8), (10), (12), if we choose , , , , and , respectively.

In [22], Qi introduced the generalized k-fractional conformable integrals which generalize the Riemann–Liouville fractional integrals [17], Hadamard fractional integrals [19], and Katugampola fractional integrals [20].

Definition 8. Let be a continuous function on a finite real interval . Then, the generalized left- and right-sided , fractional conformable integrals of order , are defined as follows:andrespectively.

Diaz and Pariguan [23] established the k-generalized gamma function , which is the generalization of the classical gamma function.

Definition 9 (See [26]). The notion of the k-gamma function is as given below:The integral from the abovementioned notion is given below as follows:with , , .

Definition 10. The beta function is defined as follows:

Definition 11. The hypergeometric function is defined as follows:

In this paper, by utilizing the Jensen–Mercer inequality, we derive the Hermite–Hadamard inequality for , convex functions via , fractional integral operators and establish some new inequalities related to the Hermite–Jensen–Mercer inequality.

3. New Hermite–Jensen–Mercer Type Inequalities via K-Fractional Integrals

Theorem 2. Let and be a , convex mapping. Then, the following fractional integral inequality holdswhere , .

Proof. Since is a , convex function, we can write it as follows:for all .
We consider and . Then, for and , we have the following:Multiplying both sides of the abovementioned equation with and then integrating w.r.t over the interval and unliving k-fractional integral operators gives the following:By setting and , we have as follows:Note thatTherefore,So, the first inequality of (21) is proved.
Now, for the proof of the second inequality of (21), we first note that is a , convex function; then, for , we can write as follows:andBy adding equations (28) and (29), we have as follows:Multiplying both sides of the abovementioned equation with and then integrating w.r.t over and utilizing k-fractional integral operator gives the following:which completes the proof of (21).

Remark 2. Theorem 2 leads to the following conclusions,(1)If we set k = 1, x = , y = , and , then Theorem 2 coincides with Theorem 2.1 of [24].(2)If we set = k = p = 1, , and , then Theorem 2 coincides with Theorem 2 of [25].(3)If we set , then Theorem 2 coincides with Theorem 2.1 of [26].

Theorem 3. Let and be a p-convex function. Then, the following fractional integral inequality holds,where , s.

Proof. It follows from the Jensen–Mercer inequality thatfor all , . Now, by a change of variable and for , and in the above inequality, we obtain the following:Multiplying both sides of the abovementioned equation with and then integrating w.r.t over the interval and utilizing the fractional integral operator gives the following:By setting and , we have as follows:By using fractional integral operators, we have as follows:which completes the proof of the first inequality of (32).
To prove the second inequality, we use the p-convexity of ; for , we obtainMultiplying both sides of the abovementioned equation by and then integrating with respect to over , we have the following:By substitution of the k-fractional integral operators as discussed above, we have the following:Adding on both sides of the abovementioned equation, we obtain as follows:Combining (37) and (41), we obtain (32).

Remark 3. By assuming , Theorem 3 reduces to the inequality (13) of Theorem 2.3 in [26].

Corollary 1. Setting = = k = p = 1 in Theorem 3 gives the following result:

Theorem 4. Let and be a p-convex function. Then, the following fractional integral inequality holds,where , s.

Proof. Since is a p-convex function, for , we have the following:for all .
Let and . Then,Putting and and solving the above inequality, we obtain the following:By using k-fractional integral operators and (26), we have the following:This completes the first part of the inequality (43). To prove the second inequality, we use the p-convexity of .Adding (48) and (49), and then using the Jensen–Mercer inequality, we obtain the following:Multiplying both sides of the abovementioned equation by and then integrating with respect to over , we have the following:Combining (48) and (51) leads to (43).

Remark 4. By assuming , Theorem 4 reduces to the inequality (14) of Theorem 2.3 in [26].

Corollary 2. By assuming p = = = k = 1, Theorem 4 reduces to the following inequality:

Lemma 1. Let , and be a differential mapping, such that . Then, we have the following:where , , and , s.

Proof. LetwhereNow, using integration by parts,Similarly, we have the following:Therefore, by substituting values of and in (54), we will obtain inequality (53).

Remark 5. We can deduce the following conclusions from Lemma 1:(1)If we substitute k = p = 1, x = , and y = , then Lemma 1 converts into Lemma 3.1 of [24].(2)If we substitute k = p = = 1, x = , and y = , then Lemma 1 converts into Lemma 3 of [27].(3)If we take p = 1, then Lemma 1 coincides with Lemma 2.5 of [26].

Theorem 5. Assume that and , and , and is a differential function, where and is p-convex, thenholds for all , .

Proof. By using Lemma 1, the Jensen–Mercer inequality, modulus property, Holder’s inequality, and p-convexity of , we have the following:Therefore, after some simple calculations, we obtain (58).

4. Conclusion

One of the most important tools to deal with the problems of pure and applied mathematics, particularly, the optimization theory is convex functions. In this paper, we studied , convex functions and established several inequalities in the setting of , fractional integral for convex functions. The inequalities established in this paper can be considered as generalizations and extensions of several existing results. It is worthy to mention here that by taking , the results of this paper reduce to the similar results for the usual fractional integral.

Data Availability

All data required for this research are included within this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this paper.

Acknowledgments

The authors are thankful to the University of Okara for providing funding for this research. This research completed under the project XJNU027 named as “The excellent young teacher of Xinjiang Normal University.” The second author is supported by the Department of Mathematics, University of Okara, Okara Pakistan.

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Copyright © 2022 Yan Dou 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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