Journal of Function Spaces
Volume 2019 (2019), Article ID 3146210, 9 pages
https://doi.org/10.1155/2019/3146210

Research Article

Hermite-Hadamard-Fejér Inequalities for Conformable Fractional Integrals via Preinvex Functions

Yousaf Khurshid,¹ Muhammad Adil Khan,¹ Yu-Ming Chu,² and Zareen Abdulhameed Khan³

¹Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
²Department of Mathematics, Huzhou University, Huzhou 313000, China
³Department of Mathematics, Princess Nora Bint Abdulrahman University, Riyadh 11538, Saudi Arabia

Correspondence should be addressed to Yu-Ming Chu; chuyuming@zjhu.edu.cn

Received 8 July 2018; Accepted 12 December 2018; Published 6 January 2019

Guest Editor: Lishan Liu

Copyright © 2019 Yousaf Khurshid 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.

Abstract.

In this paper, we present a Hermite-Hadamard-Fejér inequality for conformable fractional integrals by using symmetric preinvex functions. We also establish an identity associated with the right hand side of Hermite-Hadamard inequality for preinvex functions; then by using this identity and preinvexity of functions and some well-known inequalities, we find several new Hermite-Hadamard type inequalities for conformal fractional integrals.

1. Introduction

Let be an interval and be a convex function defined on such that , with Then the well-known (Hermite-Hadamard) inequality [1] states that

holds. If the function is concave on , then both inequalities in (1) hold in the reverse direction.

In the last few years, many researchers have shown their extensive attention on the generalizations, extensions, variations, refinements, and applications of the inequality (see [2–15]). The most well-known generalization of the inequality is the Hermite-Hadamard-Fejér inequality [16]. In 1906, Fejér [16] established the following weighted generalization of the Hermite-Hadamard inequality for symmetric functions:

for all convex functions , with and is symmetric with respect to .

It is well known that the convex sets and convex functions play important roles in the nonlinear programming and optimization theory. Many generalizations and extensions have been considered for the classical convexity in the last few decades. A significant generalization of convex functions is that of invex functions introduced by Hanson in [17]. The basic properties of the preinvex functions and their roles in optimization theory can be found in [18]. The inequalities for preinvex and log-preinvex functions were established by Noor [19, 20].

Now, we recall some notions and definitions in invexity analysis, which will be used throughout the paper (see [21, 22] and references therein).

Let be a nonempty set and the functions and be continuous.

Definition 1. The set is said to be invex with respect to (.,.) if

for all , and .

The invex set is also called a -connected set. If , then the invex set is also a convex set, but some of the invex sets are not convex [21].

Definition 2. The function is said to be preinvex with respect to on the invex set if

for all , and . The function is called preconcave if is preinvex.

The following Condition C was introduced by Mohan and Neogy [23].

Condition C. Suppose is an open invex subset of with respect to and satisfies

for any , and .

From Condition C, we clearly see that

for any , and .

The following inequality for the preinvex functions was proved by Noor [20].

Theorem 3. Let be a preinvex function on the interval (the interior of K) and , with . Then the following inequality holds:

Several important variants of inequality for preinvex functions have been provided in the literature [24]. Recently, the authors in [25] defined a new well-behaved simple fractional derivative called the “conformable fractional derivative”. Namely, the conformable fractional derivative of order at for the function is defined by

If the conformable fractional derivative of of order exists, then we say that is -fractional differentiable. The fractional derivative at is defined as .

Next, we present some basic results related to conformable fractional derivative in the following theorem.

Theorem 4 (see [25]). Let and be -differentiable at a point . Then

(i) for all .(ii) for any constant (iii) for all (iv)(v)(vi) if differentiable at

If in addition is differentiable, then

Definition 5 (see [25] conformable fractional integral). Let and . A function is -fractional integrable on if the integral

exists and is finite. All -fractional integrable functions on are indicated by

Remark 6.

where the integral is the usual Riemann improper integral and .

Recently, the conformable integrals and derivatives have been the subject of intensive research, and many remarkable properties and inequalities involving the conformable integrals and derivatives can be found in the literature [26–38].

In [39], Anderson provided the conformable integral version of inequality as follows.

Theorem 7 (see [39]). If and is an -fractional differentiable function such that is increasing, then we have the following inequality:

Moreover if the function is decreasing on , then we have

If , then this reduces to the classical inequality.

In this paper, we first establish the Hermite-Hadamard-Fejér inequality for conformable fractional integrals by using symmetric preinvex functions; then we present inequalities as their special cases (see Corollary 9). Secondly, we give an identity associated with the right side of inequality for preinvex functions using the conformable fractional integrals; then we establish inequalities for preinvex functions by use of Hölder inequality, power mean inequality, and preinvexity of functions.

2. Hermite-Hadamard-Fejér Inequalities for Conformable Fractional Integrals

The preinvex version of Fejer-Hermite-Hadamard inequality can be represented in conformable fractional integrals forms as follows.

Theorem 8. Suppose that , such that , is a preinvex function and symmetric with respect to , and is a nonnegative integrable function. Also assume that satisfies Condition C; then the inequality

holds for any .

Proof. Since is preinvex function and is symmetric with respect to , then for any , and , we have

i.e., with and , inequality (15) becomes

By change of variables, we have

So we can write

To prove the second inequality in (14), we know that is preinvex and satisfies Condition C, so we have

and similarly

Now with , we have

Also

If we add (21) and (22), we obtain

So we can write

From inequalities (18) and (24), we obtain over required result.

Corollary 9. If we put in (14), then we get

3. Type Inequalities for Conformable Fractional Integrals

Lemma 10. Let , with and be an -fractional differentiable function on for . If , then the following identity holds:

Proof. Integrating by parts, we have

where we have used the change of variable and then multiplied both sides by to get the desired result in (26).

Remark 11. If we set in (26), then we obtain the result which is proved by Barani et al. in [40]

Theorem 12. Let , such that and be an -differentiable function on for such that . If is preinvex, then we have the following inequality:

Proof. From Lemma 10, using the property of the modulus and preinvexity of , we have

Theorem 13. Let , such that and be an -differentiable function on for such that . If is preinvex for and , then we have the following inequality:

where

Proof. From Lemma 10, using the property of the modulus and preinvexity of , we have

Now by Hölder’s inequality

Similarly, we have

Hence, we have the result in (31).

Remark 14. If we set in (31), then we have the following inequality:

Theorem 15. Let , such that and be an -differentiable function on for such that . If is preinvex for and , then we have the following inequality:

where

Proof. From Lemma 10, using the property of the modulus and preinvexity of , we have

Now by the power-mean inequality

and similarly, we have

Now by the preinvexity of from above, we have

and

where we have the following:

Hence, we have the result in (37).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 11601485, and 11701176) and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant no. Y201635325).

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