Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 1470837, 7 pages
https://doi.org/10.1155/2020/1470837

Research Article

Hermite–Hadamard Inequalities for Harmonic -Convex Functions

Jian Zhong Xu,¹ Umar Raza,² Muhammad Waqas Javed,³ and Zaryab Hussain4,5

¹Department of Electronics and Information Engineering, Bozhou University, Bozhou 236800, China
²Department of Mathematics, University of Jhang, Jhang 35200, Pakistan
³Department of Sciences and Humanities, National University of Computer and Emerging Sciences (FAST), Chiniot 35400, Pakistan
⁴Department of Mathematics, Punjab College of Commerce New Campus, Faisalabad 38000, Pakistan
⁵Department of Mathematics, Government College University Faislabad, Faisalabad 38000, Pakistan

Correspondence should be addressed to Zaryab Hussain; zaryabhussain2139@gmail.com

Received 11 July 2020; Accepted 25 August 2020; Published 26 September 2020

Guest Editor: Shaohui Wang

Copyright © 2020 Jian Zhong Xu 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.

The objective of this article is to establish some Hermite–Hadamard-type inequalities via harmonic -convex functions in the framework of conformal fractional integral.

1. Introduction and Preliminaries

Let be an interval. Then, is said to be convex, if

holds and .

Let be an interval. Then, a function is said to be convex (concave), if

holds and .

It can be easily seen in [1–7] that the convex (concave) functions have extensive applications in pure and applied mathematics, and in the literature [8–15], many eminent inequalities and other properties can be found in the framework of convexity. One of the renowned inequalities in the literature of Hermite–Hadamard Integral Inequality is given below:

These both inequalities hold in reverse if the function is concave. Now, the harmonic convex set is defined as follows.

Definition 1. Let be an interval. Then, is said to be harmonic convex, if

holds with and .
Iscan [8] introduced the concept of harmonic convex function.

Definition 2. (see [8]). Let be an interval. Then, a function is called harmonic convex (concave), if

holds for all with and .
In [8], Iscan by using the concept of harmonic convex function gave a new refinement of Hermite–Hadamard inequality as

Definition 3. Let be an interval and . Then, a function is called harmonic -convex (concave), if

holds with and .
If , then harmonic -convex function becomes the classical harmonic convex function. So harmonic convex function is a special case of harmonic -convex function.
The main purpose of this article is to establish some conformable fractional estimates of Hermite–Hadamard-type inequalities via harmonic -convex functions. Before going further towards our main results, let us have a brief review of the previously well known concepts and results. These preliminaries will be highly helpful in acquiring the main results.
The eminent gamma and beta functions are defined as

The integral form of hypergeometric function is defined as

for .
Now, if with , then Riemann–Liouville integrals and of any positive order are defined as

For more details, see [11].
Recently, Abdeljawad [16] introduced the notation of right and left conformable fractional integrals for any positive order as follows.

Definition 4. (see [16]). Let . Then, the left and right conformable fractional integrals starting from of any positive order is given as

Lemma 1. (see [5]). Let f be differentiable on , and . Then,

2. Main Results

In this section, we will present our main results.

Theorem 1. Let be a harmonic (s, m)-convex function such that and . Then,

Proof. By applying the definition of harmonic (s, m)-convex function for , we have

Put and . Then,

We know that

Now, consider a function such that . Then,

Also,

So is harmonic (s, m)-convex, and also, we have

which implies that the inequality holds.

Theorem 2. Let be a harmonic (s, m)-convex function such that , where . Then,

Here, is the composition function.

Proof. From inequality 1, we have

Put and . Then,

By using change of variable technique of integration, we have

where , and

where .
Now, from (23), we have

So the inequality holds.

Theorem 3. Let be differentiable on , and . If for is harmonic -convex on , then

where

Proof. Hlder’s inequality and Lemma 1 implies that

Since is harmonically -convex on , we have

Here,

Theorem 4. Let be differentiable on , and . If for is harmonic -convex on , then

where .

Proof. Hlder’s inequality and Lemma 1 implies that

Since is harmonically -convex on , we have

where .

Theorem 5. Let be differentiable on , and . If for is harmonic -convex on , then

where

Proof. Hlder’s inequality and Lemma 1 implies that

Since is harmonically -convex on , we have

where

This completes our arguments.

3. Conclusion

In this article, we have established new Hermite–Hadamard-type inequalities via harmonic -convex functions in the framework of fractional integrals. This article may be useful for the researchers to derive some other type of inequalities.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Key Natural Science Foundation of the Education Department of Anhui Province (KJ2019A1303 and KJ2019A1300); the Key Project for Teaching Research of Bozhou University (2018zdjy01 and 2018zdjy02); and the Key Natural Science Foundation of the Education Department of Bozhou University (BYZ2018B03 and BYZ2019B03). Moreover, the second author wants to acknowledge the efforts of the vice chancellor, University of Jhang, for providing such a good environment for research.

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