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Jia, Wenyan; Yussouf, Muhammad; Farid, Ghulam; Khan, Khuram Ali

doi:https://doi.org/10.1155/2021/9945114

Mathematical Problems in Engineering

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Volume 2021 | Article ID 9945114 | https://doi.org/10.1155/2021/9945114

Hadamard and Fejér–Hadamard Inequalities for -Convex Functions via Riemann–Liouville Fractional Integrals

Wenyan Jia,¹Muhammad Yussouf,²Ghulam Farid,³and Khuram Ali Khan²

Academic Editor: Rafael Morales

Received09 Mar 2021

Revised23 Apr 2021

Accepted06 Aug 2021

Published16 Aug 2021

Abstract

In this paper, we introduce -convex function and some related functions. By applying this generalized definition, new versions of Hadamard and Fejér–Hadamard fractional integral inequalities for Riemann–Liouville fractional integrals are given. The presented results hold at the same time for different types of convexities.

1. Introduction and Preliminary Results

Convex functions play an important role in mathematical inequalities. Many inequalities for convex and related functions have been studied in recent decades and consequently published in well-reputed journals (see [1–15]). The utilization of fractional integral operators for establishing the generalized versions of classical inequalities has become a fashion in modern study of mathematical inequalities. In this regard, the Hadamard inequality is studied extensively for fractional integral operators.

The aim of this paper is to present Hadamard-type inequalities and their weighted versions called Fejér–Hadamard-type inequalities for Riemann–Liouville fractional integral operators of generalized convex functions. These fractional inequalities will hold simultaneously for convex, -convex, harmonically convex, -convex, -convex, -convex, -convex, -convex, and -convex functions.

First, we give the definition of Riemann–Liouville fractional integrals (see [16]).

Definition 1. The left and right sided Riemann–Liouville fractional integrals for a function of order are defined bywhere .

Next, we give definitions of convex and related functions which are useful for this study.

Definition 2 (see [17]). A function is said to be convex ifholds. If the inequality in (3) is reversed, then is called concave function.

Definition 3 (see [18, 19]). Let be an interval and . Then, a function is said to be -convex, ifholds for and . If the inequality in (4) is reversed, then is called -concave function.

Definition 4 (see [3]). Let be a positive function. Let be an interval and . A function is said to be -convex, ifholds for and . If the inequality in (5) is reversed, then is called -concave function.

The following definition unifies several kinds of convex functions for example -convex, -convex, and -convex functions in a single inequality.

Definition 5 (see [20]). Let be an interval containing and let be a non-negative function. A function is called -convex function, if is non-negative and for all and , one has

In the upcoming section, we define a new generalized notion which will be called -convex function. It will generate all kinds of convex functions discussed in aforementioned premise. By using this new definition and Riemann–Liouville fractional integrals, we will prove two versions of the Hadamard inequality. In Section 3, we give the weighted versions of the Hadamard inequalities presented in Section 2. Also, results for -convex, -convex, and -convex functions are explicitly given in the form of corollaries.

2. Hadamard-Type Inequalities for --Convex Function

We define --convex function as follows.

Definition 6. Let be an interval containing and let be a non-negative function. Let be an interval and . A function is said to be --convex, ifholds provided for and .

For different choices of , and , the outcomes of Definition 6 are given in the remark as follows.

Remark 1. (i)Fixing in Definition 6, it gives -convexity (see 20, [Definition 4.5]).(ii)Fixing and in Definition 6, it gives -HA-convexity (see 4, Definition 2.1]).(iii)Fixing in Definition 6, it gives -convex function (see [3]).(iv)Fixing and in Definition 6, it gives -convex function (see [21]).(v)Fixing and in Definition 6, it gives -convex function (see [17]).

We define -convex function by setting in Definition 6 as follows.

Definition 7. Let be an interval containing and let be a non-negative function. Let be an interval and . A function is said to be -convex, ifholds provided for and .

We define -convex function by setting in Definition 6 as follows.

Definition 8. Let be an interval and . A function is said to be -convex, ifholds provided for and .

We define -convex function by setting in Definition 6 as follows.

Definition 9. Let be an interval containing and let be a non-negative function. Let be an interval and . A function is said to be -convex, ifholds for , .

We define -HA-convex function by setting in Definition 6 as follows.

Definition 10. Let be an interval containing and let be a non-negative function. Let be an interval and . A function is said to be -HA-convex, ifholds for , , , and .

We define -convex function by setting and in Definition 6 as follows.

Definition 11. Let be an interval containing and let be a non-negative function. Let be an interval. A function is said to be -convex, ifholds provided for and .

We define -Godunova–Levin function by setting and in Definition 6 as follows.

Definition 12. Let be an interval containing and let be a non-negative function. Let be an interval and . A function is said to be -Godunova–Levin, ifholds provided for and .

In the next results, it is assumed that functions , and are integrable and all involved integrals are finite.

Theorem 1. Let be a positive -convex function as given in Definition 6 with . Then, the following inequality for fractional integral operators (1) and (2) holds:

Proof. By -convexity of , one can haveBy setting and in (15), we getMultiplying both sides of (16) by and integrating over , we getSetting , that is, and , that is, , on the right hand side of (17) and after simplification, one can get first inequality of (14).
Again by using the -convexity of , the right hand side of (16) leads to the following inequality:Multiplying on both sides of (18) and integrating over , we haveSetting , that is, and , that is, , in (19) and after calculation, one can get the second inequality of (14).

The inequality established in (14) gives the Hadamard inequality for -convex function by fixing . The readers can also obtain the Hadamard inequality for -convex, -convex, -convex, and all deducible functions by fixing parameters and the function of their choice. Also, inequality (14) provides fractional versions of the Hadamard inequality for Riemann–Liouville fractional integrals of functions stated in the introduction section.

Further consequences of the above theorem are stated in the following.

Remark 2. (i)On fixing , , and , [[14], Theorem 2] is obtained.(ii)On fixing , , , and , classical Hadamard inequality [22] is obtained.(iii)On fixing , , and , [[23], Theorem 2.4] is obtained.

Now, we give another variant of the Hadamard inequality as follows.

Theorem 2. Let the assumptions of Theorem 1 hold. Then, we have the following inequality:

Proof. By setting and in (15), we getMultiplying both sides of (21) by and integrating over , we getSetting , that is, and , that is, , on the right hand side of (22) and after simplification, one can get first inequality of (20).
Again by using -convexity of , the right hand side of (21) leads to the following inequality:Multiplying on both sides of (23) and integrating over , we haveSetting , that is, and , that is, , in (24) and after calculation, one can get second inequality of (20).

The above version of the Hadamard inequality gives the corresponding inequalities for -convex, -convex, -convex, and all deducible functions by fixing parameters and the function .

Further consequences of the above theorem are stated in the following.

Remark 3. (i)On fixing , , and in Theorem 2, [[24], Theorem 2.1 (i)] is obtained.(ii)On fixing , , and in Theorem 2, [[24], Theorem 2.1 (ii)] is obtained.(iii)On fixing , , and in Theorem 2, [[24], Corollary 2.1] is obtained.

Remark 4. From Theorems 1 and 2, one can deduce results for convex, -convex, -convex, -convex, -convex, -convex, -convex, and -convex functions.

3. Fejér–Hadamard-Type Inequalities

In this section, we present the Fejér–Hadamard-type inequalities for -convex functions by applying definition of Riemann–Liouville fractional integral operators. We also give results for new definitions obtained in Section 2.

Theorem 3. Let be a positive -convex function as given in Definition 6 and . If is a positive function, then the following inequality for fractional integral operators (1) and (2) holds:

Proof. Multiplying (16) by and integrating over , we getSetting , that is, , in (26) and utilizing condition , by fractional integral operators (1) and (2), one can get first inequality of (25).
Now, multiplying in (18) and integrating over , we haveAgain setting , that is, , in (27) and utilizing condition , by using definitions of fractional integral operators (1) and (2), one can get second inequality of (25).

The consequences of the above theorem are stated in the following.

Remark 5. (i)On fixing , , , and , [[14], Theorem 2] is obtained.(ii)On fixing , , , , and , the classical Hadamard inequality [25] is obtained.(iii)On fixing , , , and , the Fejér–Hadamard inequality [26] is obtained.

Now, we give another variant of the Fejér–Hadamard inequality as follows.

Theorem 4. Let the assumptions of Theorem 3 hold. Then, we have the following inequality:

Proof. Multiplying (21) by and integrating over , we getSetting , that is, , in (29) and utilizing condition , by fractional integral operators (1) and (2), one can get first inequality of (28).
Multiplying on both sides of (23) and integrating over , we haveAgain setting , that is, , in (30) and utilizing condition , by fractional integral operators (1) and (2), one can get second inequality of (28).

The consequences of the above theorem are stated in the following.

Remark 6. (i)On fixing , , , and in Theorem 4, [[24], Theorem 2.1 (i)] is obtained.(ii)On fixing , , , and in Theorem 4, [[24], Theorem 2.1 (ii)] is obtained.(iii)On fixing , , , and in Theorem 4, [[24], Corollary 2.1] is obtained.

Remark 7. From Theorems 3 and 4, one can deduce results for convex, -convex, -convex, -convex, -convex, -convex, and -convex functions.

3.1. Results for -Convex Functions

By setting in Theorems 1–4, the results for -convex functions are obtained as follows.

Theorem 5. Under the assumptions of Theorem 1, the following inequality holds for -convex functions:

Theorem 6. Under the assumptions of Theorem 2, the following inequality holds for -convex functions:

Theorem 7. Under the assumptions of Theorem 3, the following inequality holds for -convex functions:

Theorem 8. Under the assumptions of Theorem 4, the following inequality holds for -convex functions:

3.2. Results for -Convex Functions

By setting in Theorems 1, 2, 3, and 4, the results for -convex functions are obtained as follows.

Theorem 9. Under the assumptions of Theorem 1, the following inequality holds for -convex functions:

Theorem 10. Under the assumptions of Theorem 2, the following inequality holds for -convex functions:

Theorem 11. Under the assumptions of Theorem 3, the following inequality holds for -convex functions:

Theorem 12. Under the assumptions of Theorem 4, the following inequality holds for -convex functions:

3.3. Results for -Convex Functions

By setting in Theorems 1, 2, 3, and 4, the results for -convex functions are obtained as follows.

Theorem 13. Under the assumptions of Theorem 1, the following inequality holds for -convex functions:

Theorem 14. Under the assumptions of Theorem 2, the following inequality holds for -convex functions:

Theorem 15. Under the assumptions of Theorem 3, the following inequality holds for -convex functions:

Theorem 16. Under the assumptions of Theorem 4, the following inequality holds for -convex functions:

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Wenyan Jia proved the main results, Muhammad Yussouf wrote the paper, Ghulam Farid supervised this work, and Khuram Ali Khan verified the results.

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Copyright © 2021 Wenyan Jia 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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