Generalization of Hermite-Hadamard Type Inequalities via Conformable Fractional Integrals

Abstract

We establish a Hermite-Hadamard type identity and several new Hermite-Hadamard type inequalities for conformable fractional integrals and present their applications to special bivariate means.

1. Introduction

In the field of nonlinear programming and optimization theory, no one can ignore the role of convex sets and convex functions. For the class of convex functions, many inequalities have been introduced such as Jensen’s, Hermite-Hadmard, and Slater’s inequalities. Among those inequalities, the most famous and important inequality is the Hermite-Hadamard’s inequality [1] which can be stated as follows.

Let be an interval and be a convex function defined on . Then the double inequalityholds for all with . Both inequalities in (1) hold in the reverse direction if the function is concave on .

In the last 60 years, many efforts have gone on generalizations, extensions, variants, and applications for the Hermite-Hadamard’s inequality (see [2–13]). Anderson [14] and Sarikaya et al. [15] provide the important variants for the Hermite-Hadamard’s inequality.

Recently, the author in [16] gave a new definition for the (conformable) fractional derivative as follows.

Let and be a real-valued function. Then the -order (conformable) fractional derivative of at is defined by is said to be -differentiable if the -order (conformable) fractional derivative of exists, and the -order (conformable) fractional derivative of at is defined as .

Now we discuss some theorems for the (conformable) fractional derivative.

Theorem 1. Let and be -differentiable at . Then one has the following:(i) for all .(ii) for all constants .(iii) for all constants .(iv).(v)(vi) if is differentiable at . In addition,if is differentiable.

Definition 2 (conformable fractional integral). Let and . Then the function is said to be -fractional integrable on if the integralexists and is finite. All -fractional integrable functions on are indicated by

Remark 3.  where the integral is the usual Riemann improper integral and

Recently, the conformable integrals and derivatives have attracted the attention of many researchers, and many remarkable properties and inequalities for the conformable integrals and derivatives can be found in the literature [17–24]. Anderson [14] found the conformable integral version of the Hermite-Hadamard inequality as follows.

Theorem 4 (see [14]). 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 haveIf , then inequalities (6) and (7) reduce to the classical Hermite-Hadamard’s inequalities.

The main purpose of the article is to present the conformable fractional integrals version of the Hermite-Hadamard’s inequality. We first establish an identity for the conformable fractional integrals (Lemma 5) and discuss their special cases. Then applying Jensen’s inequality, power mean inequality, Hölder inequality, the convexity of the functions and , and the identity given by Lemma 5, we obtain inequalities for conformable fractional integrals version of the Hermite-Hadamard’s inequality. At last, using particular classes of convex functions we find several new inequalities for some special bivariate means. For some related results, see [25, 26].

2. Main Results

The main results of our work can be calculated with the help of the following lemma associated with inequality (8).

Lemma 5. Let with , , and be an -fractional differentiable function on . Then the identityholds for any if .

Proof. It follows from Theorem 1, Definition 2, and integrating by parts that where we have used the changes of variable and to get the desired result.

Remark 6. Let . Then identity (8) becomes which was proved by Kavurmaci et al. in [2].

Theorem 7. Let with , , and be an -differentiable function on . Then the inequality holds for any if and is convex on .

Proof. Let , , and . Then we clearly see that both the functions and are convex. From Lemma 5 and the convexity of , , and , we haveFrom the final upper bound above, we have the following:

Remark 8. Let . Then inequality (12) leads to which was proved by Kavurmaci et al. in [2].

Theorem 9. Let with , , such that and be an -differentiable function on . Then the inequalityholds for any if and is convex on , where

Proof. It follows from inequality (13) that Making use of Hölder’s inequality, one hasSimilarly, we have Hence, we have the result in (16).

Remark 10. By putting in (16), we obtain the inequality which was proved by Kavurmaci et al. in [2].

Theorem 11. Let with , , and be an -differentiable function on . Then the inequalityholds for any if and is convex on , where

Proof. It follows from inequality (13) that Making use of the power mean inequality, we get Similarly, we have From the convexity of , we have and where we have also used the facts that Hence, we have the result in (22).

Remark 12. If , then inequality (22) becomes which can be found in [2].

Theorem 13. Let with , , and be an -differentiable function on . Then the inequalityis valid for any if and is convex on , where

Proof. From Theorem 1, Definition 2, and Lemma 5, we get Making use of power mean inequality, we get Similarly, we have It follows from the convexity of that andwhere we have used the identities Hence, we have the result in (31).

Theorem 14. Let with , and be an -differentiable function on . Then the inequalityholds for any if and is concave on for some , where

Proof. We clearly see that is concave because is concave for some (see [27]). From Theorem 1, Definition 2, Lemma 5, the concavity of , and Jensen’s integral inequality, we have where we have used the identities and

Remark 15. If , then inequality (39) becomes

3. Applications to Special Means

A bivariate function is said to be a mean if for all . Recently, the mean value theory has been the subject of intensive research, and many remarkable inequalities and properties for various bivariate means can be found in the literature [28–33].

In this section, we use the results obtained in Section 2 to give some applications to the weighted arithmetic mean and -th generalized logarithmic mean

Proposition 16. Let with and . Then the inequality holds for any and .

Proof. Let . Then the result follows easily from Theorem 7 and the convexity of on the interval .

Proposition 17. Let with and . Then the inequality holds for any and .

Proof. Let . Then Proposition 17 follows from Theorem 7 and the convexity of the function on the interval .

Proposition 18. Let with and . Then the inequality holds for all and , where and are defined as in Theorem 11.

Proof. Let . Then Proposition 18 follows from Theorem 11 and the convexity of on immediately.

Proposition 19. Let with and . Then the inequality holds for all and , where and are defined as in Theorem 11.

Proof. Let . Then the result follows easily from Theorem 11 and the convexity of on the interval .

4. Conclusion

In the article, we establish an identity and several new inequalities of Hermite-Hadamard type for conformable fractional integrals by use of the convexity theory and Jesen’s inequality, Hölder inequality, and power mean inequality and present their applications to special bivariate means. The given Hermite-Hadamard type inequalities for conformable fractional integrals are the generalizations of the corresponding results established by Kavurmaci, Avci, and Özdemir in [2], and the idea may stimulate further research in the theory of Hermite-Hadamard’s inequalities, conformable fractional integrals, and generalized convex functions.

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 (Grant nos. 61673169, 11701176, 11626101, and 11601485) and the Science and Technology Research Program of Zhejiang Educational Committee (Grant no. Y201635325).