Abstract

In this article, we establish some new Hermite–Hadamard-type inequalities involving the conformable fractional integrals. As applications, several inequalities for the approximation error in the midpoint formula and certain bivariate means are derived.

1. Introduction

Let be an interval. Then, a real-valued function is said to be convex (concave) on I if the inequalityholds for all with and .

It is well known that the convex functions have wide applications in pure and applied mathematics [1–27]. In particular, many remarkable inequalities and properties for the convex functions can be found in the literature [28–64]. Recently, a great deal of generalizations, extensions, and variants have been made for the convexity [65–72].

Let with and be a convex function. Then, the classical Hermite–Hadamard inequality [73–77] states that the double inequalityholds.

Alomari et al. [78] and Chen and Feng [79] proved the following.

Theorem 1 (see [78]). Let be an interval, be the interior of I, and be a differentiable mapping. Then, the identityholds for with if .

Theorem 2 (see [79]). Let be an interval, be the interior of I, and be a differentiable mapping. Then, the identityholds for with and if .

Now, we recall the definitions of the conformable fractional derivative and integral.

Definition 1 (see [80]). Let be a real-valued function and . Then, the α-order conformable fractional derivative of ϕ at is defined bywhere ϕ is said to be α-differentiable at z if exists. In particular, is defined byand we use or to denote .
The following formulas can be derived from Definition 1 immediately.

Theorem 3. Let and and be two α-differentiable real-valued functions at . Then, one hasfor any constants . If is differentiable at , then

Moreover, if is differentiable, then

Definition 2. Let and with . Then, the real-valued function is said to be α-fractional integrable if the integralexists and is finite. All α-fractional integrable functions on are denoted by .

Remark 1. where the integral is the usual Riemann improper integral and .
Anderson [81] established the Hermite–Hadamard inequality for the conformable fractional integral as follows.

Theorem 4. Let , with , and be an α-fractional differentiable function such that is increasing. Then, one has

Moreover, if the function ϕ is decreasing on , then

Remark 2. If , then we clearly see that inequalities (12) and (13) reduce to inequality (2).
The aim of this article is to establish some new Hermite–Hadamard-type inequalities for the conformable fractional integrals and present their applications in the error estimations of the midpoint formula and certain bivariate means.

2. Main Results

In order to establish our main results, we need a lemma which we present in this section.

Lemma 1. Let , with , and be an α-fractional differentiable function. Then, the identityholds for any if .

Proof. Integrating by parts, we havewhere we use the changes of variables in and in . Then, multiply by and by to get the desired result.

Remark 3. Let , then identity (14) reduces to (4).

Remark 4. Let and , then identity (14) reduces to (3).

Theorem 5. Let , with , and be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequalityholds for any if and is convex on , where

Proof. It follows from Lemma 1 and the convexity of the functions and for together with that

Remark 5. Let . Then, inequality (16) leads towhere , , , and are defined as in Theorem 5.

Theorem 6. Let , with , and such that and be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequalityis valid for all if and is convex on , where

Proof. Making use of Lemma 1, the convexity of on , Hölder inequality, and the property of the modulus, we get

Remark 6. Let . Then, inequality (20) leads towhere , , , and are defined as in Theorem 6.

Theorem 7. Let , with , , and be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequalitytakes place for all if and is convex on , where

Proof. It follows from Lemma 1, convexity of the function , power mean inequality, and the property of the modulus that

Remark 7. Let . Then, from inequality (24), we getwhere , , , , , and are defined as in Theorem 7.

Theorem 8. Let , with , , and be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequalitytakes place for all if and is convex on , where

Proof. From Lemma 1 and convexity of , we get

Remark 8. Let . Then, inequality (28) becomeswhere , , , , , and are defined as in Theorem 8.

Theorem 9. Let , with , , and be an α-fractional differentiable function. Then, the Hermite–Hadamard-type inequalityholds for any if and is concave on , where

Proof. Let . Then, the power mean inequality and the concavity of lead to the conclusion thatwhich shows that is also concave.
Making use of Lemma 1, we getApplying Jensen integral inequality, we haveSimilarly, one has

Remark 9. Let . Then, inequality (32) giveswhere , , , and are defined as in Theorem 9.

Theorem 10. Let , with , and be an α-fractional differentiable function. Then, the Hermite-Hadamard-type inequalityholds for all if and is convex on , where , , , and are defined as in Theorem 5.

Proof. It follows from Lemma 1 and the convexity of that

Remark 10. Let . Then, inequality (39) leads to

3. Applications to Bivariate Means

A real-valued function is said to be a bivariate mean [82] if for all . Recently, the properties and applications for the bivariate means and their related special functions have attracted the attention of many researchers [83–93].

Let with , , and . Then, the arithmetic mean , logarithmic mean , and -th generalized logarithmic mean are defined byrespectively.

In this section, we shall present several inequalities involving the arithmetic, logarithmic, and generalized logarithmic means.

Let and . Then, Theorems 5 and 6 lead to Theorems 11 and 12 immediately.

Theorem 11. Let with , , and . Then, the inequalitieshold for all , where , , , and are defined as in Theorem 5.

Theorem 12. Let with , , , and with . Then, the inequalitieshold for all , where and are defined as in Theorem 6.

Let and . Then, Theorem 6 leads to Theorems 13 immediately.

Theorem 13. Let with , , and . Then, the inequalitieshold, where , , , , , and are defined as in Theorem 8.

4. Applications to Midpoint Formula

Let P be the partition of the points of the interval and consider the quadrature formulawhereis the midpoint version and denotes the associated approximation error. In this section, we are going to derive some new estimates for the midpoint formula.

Theorem 14. Let with , , and be an α-differentiable function. Then, one hasif and is convex on , where

Proof. Applying Theorem 5 on the subintervals of the partition P, we haveMaking use of Theorems 6–8 and similar arguments as in the proof of Theorem 14, we get Theorems 15 and 16 immediately.

Theorem 15. Let with , , and be an α-differentiable function and with . Then, the inequalityholds for all if and is convex on , where

Theorem 16. Let with , , and be an α-differentiable function and . Then, the inequalityholds for all if and is convex on , where

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.

Authors’ Contributions

All authors contributed equally to writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of China (grant nos. 61673169, 11971142, 11871202, 11301127, 11701176, 11626101, and 11601485).