Abstract

Due to applications in almost every area of mathematics, the theory of convex and nonconvex functions becomes a hot area of research for many mathematicians. In the present research, we generalize the Hermite–Hadamard-type inequalities for -convex functions. Moreover, we establish some new inequalities via the Caputo-Fabrizio fractional integral operator for -convex functions. Finally, the applications of our main findings are also given.

1. Introduction

In the last few decades, the subject of fractional calculus got attention of many researchers of different fields of pure and applied mathematics like mechanics, convex analysis, and relativity [1–3]. Nowadays, the researches in convex analysis cannot ignore the deep connectivity of both inequalities in convex analysis and fractional integral operator. Niels Henrik Abel gave birth to fractional calculus. The applications of fractional calculus can be seen in [4–9]. The first appearance of fractional derivative had been seen in a letter. The letter was written to Guillaume de lHopital by Gottfried Wilhelm Leibniz in 1695.

The fractional calculus techniques can be seen in many branches of science and engineering. Geometric and physical interpretation of fractional integration and fractional differentiation can be viewed in [10]. There are different fractional integral operators in which we use the integral inequalities (see, for example, [11–16]). The well-known inequality given by Hermite in 1881 can be stated as follows.

Theorem 1. Let be a -convex function defined on the interval of real numbers and with . Then, the following inequality holds:

The fractional Hermite–Hadamard and Hermite–Hadamard inequalities via fractional integral can be seen in [17, 18]. For the history of Hermite–Hadamard-type inequalities, we refer to the readers [19, 20]. The outstanding applications of fractional calculus and fractional derivatives and integrals are given in [21]. Moreover, we refer the readers for a detailed study [22–27].

In the present article, we generalize the Hermite–Hadamard-type inequalities for -convex functions. Moreover, we establish some new inequalities via the Caputo-Fabrizio fractional integral operator for -convex functions. Finally, the applications of our main findings are also given.

This paper is organized as follows: in Section 2, some preliminaries are given. In Section 3, we generalized Hermite–Hadamard via Caputo-Fabrizio for -convex functions. In Section 4, we give some results related to Caputo-Fabrizio, and in Section 5, some applications to special means are given.

2. Preliminaries

We will start with some basic definitions related to our work.

Definition 2 (convex function) [28]. Let be an extended real-valued function defined on a convex set . Then, the function is convex on if for all and .

Definition 3 (-convex function) [29]. A function is said to be a -convex function if

Definition 4 (-convex function) [30]. Let be a nonnegative function. We say that is an -convex function or that , if is nonnegative, and for all , we have

If inequality (4) is reversed, then is said to be -convex, i.e., .

Remark 5. (1)If we take , then (4) reduces to (2)(2)If the function has the property: for all , then any nonnegative convex function belongs to the class (3)If the function has the property: for all , then any nonnegative convex function belongs to the class

Definition 6 (-convex function) [31]. A function is called a -convex function if

Definition 7 (Caputo-Fabrizio fractional time derivative) [32]. The usual Caputo fractional time derivative of order is given by with and . By changing the Kernal with the function and with , we obtain the new definition of fractional time derivative:

Definition 8 (Caputo-Fabrizio fractional integral) [32]. Let ; then, the definition of the left fractional derivative in the sense of Caputo and Fabrizio becomes and the associated fractional integral is where is a normalization function satisfying .
In the right case, we have and the associated fractional integral is

Lemma 9. Let be a differentiable mapping on with . If , then the following inequality holds: where .

3. A Generalized Hermite–Hadamard-Type Inequality via the Caputo-Fabrizio Fractional Operator for a -Convex Function

The double inequality named as Hermite–Hadamard inequality is considered one of the fundamental inequalities for convex functions.

Theorem 10. Let a function be a -convex function on and if ; then, the following double inequality holds: where

Proof. Let be a -convex function; then, Hermite–Hadamard for a -convex function is as follows: Since is a -convex function on , we can write Multiplying both sides of (15) with and adding , we have For the proof of the right hand side of the Hermite–Hadamard-type inequality, we have Multiplying both sides of (17) with and adding , we get By recognizing (18), the proof of the right hand side of (13) is completed. This completes the proof.

Remark 11. If we put and , then we will get the Hermite-Hadamard inequality for convex function.

Theorem 12. Let be a -convex function. If , then we have the following inequality: with and , where , , and is a normalization function.

Proof. Since and are -convex functions on , we have Multiplying both sides of the above inequalities, we have Integrating with respect to over and making the change of variable, we obtain which implies By multiplying both sides with and adding , we have Thus, and with suitable rearrangements, the proof is completed.

Remark 13. If we put and in the above theorem, we get the results for the classical convex function.

Theorem 14. Let be a -convex function. If , then we have the following inequality: with and , where , , and is a normalization function.

Proof. Since for , we have Multiplying (2) and (28), we have Integrating the above inequality with respect to over and making the change of variable, one obtains By multiplying both sides with and subtracting , we have By multiplying both sides of the above inequality by , we get the required inequality (28).

Remark 15. If we put and in the above theorem, then we will get the result for the convex function.

In this section, firstly we generalize a lemma; then, we prove our main theorem with the help of this lemma.

Lemma 16. Let be a differentiable mapping on with . If and , the following equality holds: where and is a normalization function with .

Proof. From Lemma 9, we can observe that By multiplying and subtracting , we have This completes the proof.

Theorem 17. Let be a differential mapping on and be -convex on where with . If and , the following inequality holds: where where and is a normalization function.

Proof. This completes the proof.

Theorem 18. Let be a differential mapping on and be -convex on where with , , and . If and , the following inequality holds: where where and is a normalization function.

Proof. By a similar argument to the proof of the previous theorem, by using lemma, the Hölder inequality, and convexity of , we get This completes the proof.

5. Application to Special Means

The applications to special means are also used to confirm the accuracy of the findings for real numbers such that .

The arithmetic mean of two numbers and is defined as

The generalized logarithmic mean is defined as

Now, using the results in Section 4, we have some applications to the special means of real numbers.

Proposition 19. Let ; then,

Proof. In the inequality proven in Theorem 17, if we set with and , then we obtain the result immediately.

Remark 20. If we put and in this proposition, we will obtain this result for the convex function.

Proposition 21. Let ; then,

Proof. In the inequality proven in Theorem 17, if we set where is an even number with and , then we obtain the result immediately.

Remark 22. If we put and in this preposition, we will obtain this result for the convex function.

6. Conclusion

Convexity is very important for solving optimization problems. Fractional calculus together with convexity plays an important role in solving real-life problems. In this paper, we established several Hermite–Hadamard-type inequalities in the setting of a fractional integral operator for -convex functions. We also presented some applications in means. Our results generalized several existing results.

Conflicts of Interest

The authors do not have any competing interests.

Authors’ Contributions

Dong Zhang proved the main results, Muhammad Shoaib Saleem proposed the problem and supervised this work, Thongchai Botmart analyzed the results and arranged the funding for this paper, M. S. Zahoor proved the main results, and R. Bano wrote the first version of this paper.