#### Abstract

The aim of this paper is to present the fractional Hadamard and Fejér-Hadamard inequalities for exponentially -convex functions. To establish these inequalities, we will utilize generalized fractional integral operators containing the Mittag-Leffler function in their kernels via a monotone function. The presented results in particular contain a number of fractional Hadamard and Fejér-Hadamard inequalities for -convex, -convex, -convex, exponentially convex, exponentially -convex, and convex functions.

#### 1. Introduction and Preliminaries

Integral operators play an important role in the subject of mathematical analysis. Fractional integral operators have been proven very useful in almost all fields of science and engineering. By using fractional integral operators, a lot of well-known inequalities have been studied, and in the consequent, they are generalized and extended in the subject of fractional calculus. Fractional integral inequalities provide support in the formation of modeling of physical phenomenons. They also provide their role in the uniqueness of solutions of fractional boundary value problems. A large number of fractional integral inequalities exist in literature due to fractional integral operators (see, [15]). The Riemann-Liouville fractional integral operators are the first formulation of fractional integral operators of nonintegral order.

Definition 1 [6]. Let . The Riemann-Liouville fractional integral operators and of order are defined by where .

Next, we give the definition of generalized fractional integral operators containing the Mittag-Leffler function in their kernels as follows.

Definition 2 [7]. Let , , with , , and . Let and . Then, the generalized fractional integral operators and are defined by where is the generalized Mittfag-Leffler function defined as follows:

Lemma 3 [7]. If , , with , and , then

Recently in [8], Farid defined the following unified integral operators.

Definition 4. Let , be the functions such that be a positive and integrable and be a differentiable and strictly increasing. Also, let be an increasing function on and , , and . Then, for , the integral operators and are defined by

If we take in (7) and (9), then we get the following generalized fractional integral operators containing the Mittag-Leffler function.

Definition 5. Let , be the functions such that be a positive and integrable and be a differentiable and strictly increasing. Also, let , , and . Then, for , the integral operators and are defined by

Remark 6. The operators (9) and (10) are the generalization of the following fractional integral operators: (i)By taking and , the Riemann-Liouville fractional integral operators (1) and (3) can be achieved(ii)By taking , the fractional integral operators (5) and (6) can be achieved(iii)By taking and , the fractional integral operators defined by Salim-Faraj in [9] can be achieved(iv)By taking and , the fractional integral operators defined by Rahman et al. in [10] can be achieved(v)By taking , , and , the fractional integral operators defined by Srivastava-Tomovski in [11] can be achieved(vi)By taking , , and , the fractional integral operators defined by Prabhakar in [12] can be achieved

From generalized fractional integral operator (9), we have

Hence,

Similarly, from the generalized fractional integral operator (10), we have

In this paper, we will use the following notations frequently:

Convex functions are also very important in the field of mathematical analysis (see, [1315]).

A real-valued function is said to be convex on ( is an interval in ), if holds, for all , and . The function is said to be concave if the reverse of inequality (16) holds.

Convex functions are further extended and generalized in various ways by using different techniques. For example in [16], Qiang et al. introduced exponentially -convex function which is a generalization of -convex, -convex, -convex, exponentially convex, exponentially -convex, and convex functions.

Recently, is as follows.

Definition 7. Let and be an interval. A function is said to be exponentially -convex on , if holds, for all , , and .

Remark 8. (i)If we take in (17), then exponentially -convex function defined by Mehreen and Anwar in [17] can be achieved(ii)If we take in (17), then exponentially convex function defined by Awan et al. in [18] can be achieved(iii)If we take in (17), then -convex function defined by Efthekhari in [19] can be achieved(iv)If we take and in (17), then -convex function defined by Hudzik and Maligranda in [20] can be achieved(v)If we take and in (17), then -convex function defined by Toader in [21] can be achieved(vi)If we take and in (17), then convex function (16) can be achieved

A convex function is also equally defined by the well-known Hadamard inequality stated as follows: where is a convex function.

In [22], Fejér gave a generalization of the Hadamard inequality stated as follows: where is a convex function and is positive, integrable, and symmetric to .

The inequality (19) is well known as the Fejér-Hadamard inequality. The Hadamard and the Fejér-Hadamard inequalities have been analyzed by many authors and produced frequently their generalizations, refinements, and extensions (see, [2337]).

Here, we will give two versions of the generalized fractional Hadamard inequality.

Theorem 9. Let , be the real-valued functions. If be a integrable and exponentially -convex and be a differentiable and strictly increasing. Then, for integral operators (9) and (10), the following inequalities hold: where for , for and .

Proof. Since is exponentially -convex function on , for , we have the following inequalities:

Multiplying both sides of (21) with and integrating over , we have

Putting and in (23), we get

By using (9), (10), and (14), the first inequality of (20) is obtained.

Now, multiplying both sides of (22) with and integrating over , we have

First, we calculate the following integrals of (25): as for . Hence,

By using (9), we have

Now, putting , , and using (27), (28) in (25), the second inequality of (20) is obtained.

Special cases of the above generalized fractional Hadamard inequality are highlighted in the following remark.

Remark 10. (i)If we take in (20), then Hadamard inequality for the exponentially -convex function of second kind is obtained(ii)If we take in (20), then Hadamard inequality for the exponentially convex function is obtained(iii)If we take , , and in (20), then [35] (Theorem 2.1) is obtained(iv)If we take , , and in (20), then [35] (Theorem 3.1) is obtained(v)If we take , , , and in (20), then [27] (Theorem 2.1) is obtained(vi)If we take , , , and in (20), then [28] (Theorem 3) is obtained(vii)If we take , , , and in (20), then [36] (Theorem 2) is obtained(viii)If we take , , , and in (20), then [30] (Theorem 2.1) is obtained(ix)If we take , , , , and in (20), then classical Hadamard inequality is obtained

In the following, we give another version of the Hadamard inequality for generalized fractional integral operators.

Theorem 11. Let , be the real-valued functions. If be a integrable and -convex and be a differentiable and strictly increasing. Then, for integral operators (9) and (10), the following inequalities hold: where for , for and .

Proof. Since is exponentially -convex function on , for , we have the following inequalities:

Multiplying both sides of (30) with and integrating over , we have

Putting and in (32), we get

By using (9), (10), and (14), the first inequality of (29) is obtained.

Now, multiplying both sides of (31) with and integrating over , we have

Putting and in (34), then by using (9), (10), and (27), the second inequality of (29) is obtained.

The following remark establishes the connection with already published results.

Remark 12. (i)If we take in (29), then Hadamard inequality for the exponentially -convex function of second kind is obtained(ii)If we take in (29), then Hadamard inequality for the exponentially convex function is obtained(iii)If we take , , and in (29), then [32] (Theorem 2.11) is obtained(iv)If we take , , , and in (29), then [29] (Theorem 3.10) is obtained(v)If we take , , , and in (29), then [37] (Theorem 4) is obtained(vi)If we take , , , , and in (29), then classical Hadamard inequality is obtained