Hermite–Hadamard Inequalities for Harmonic -Convex Functions

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, ifholds and .

Let be an interval. Then, a function is said to be convex (concave), ifholds 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, ifholds 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), ifholds 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), ifholds 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 asThe integral form of hypergeometric function is defined asfor .
Now, if with , then Riemann–Liouville integrals and of any positive order are defined asFor 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 havePut and . Then,We know thatNow, consider a function such that . Then,Also,So is harmonic (s, m)-convex, and also, we havewhich 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 havePut and . Then,By using change of variable technique of integration, we havewhere , andwhere .
Now, from (23), we haveSo the inequality holds.

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

Proof. Hlder’s inequality and Lemma 1 implies thatSince is harmonically -convex on , we haveHere,

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

Proof. Hlder’s inequality and Lemma 1 implies thatSince is harmonically -convex on , we havewhere .

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

Proof. Hlder’s inequality and Lemma 1 implies thatSince is harmonically -convex on , we havewhereThis 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.