New Hadamard and Fejér-Hadamard fractional inequalities for exponentially m-convex function

In this article, we present new fractional Hadamard and Fejér-Hadamard inequalities for generalized fractional integral operators containing Mittag-Leffler function via a monotone function. To establish these inequalities we will use exponentially m-convex functions. The presented results in particular contain a number of fractional Hadamard and Fejér-Hadamard inequalities for functions deducible from exponentially m-convex functions.

From generalized fractional integral operator (9), we have and similarly, from generalized fractional integral operator (10), we get We will use the following notations in the article: In [27], Mehmood et al., proved the following Hadamard and Fejér-Hadamard inequalities for exponentially m-convex functions via generalized fractional integral operators (5) and (6).
If η is exponentially m-convex function, then the following inequalities hold: If η is exponentially m-convex function, then the following inequalities hold: → R be a function which is non-negative and integrable. If η is exponentially m-convex function and η(v) = η(ς 1 + mς 2 − mv), then the following inequalities hold: In this article, we establish the Hadamard and the Fejér-Hadamard inequalities for exponentially m-convex functions by the generalized fractional integral operators (9) and (10) containing Mittag-Leffler function via a monotone function. These inequalities lead to produce results for generalized fractional integral operators given in Remark 1. In Section 2, we prove the Hadamard inequalities for generalized fractional integral operators (9) and (10) via exponentially m-convex functions. In Section 3, we prove the Fejér-Hadamard inequalities for these generalized fractional integral operators via exponentially m-convex functions. In whole paper, we will consider real parameters of the Mittag-Leffler function.

Corollary 1.
Under the assumptions of Theorem 8 if we take m = 1, then we get following inequalities for exponentially convex function:

Remark 2.
1. If we take κ(u) = u in (13), then Theorem 5 is obtained. 2. If we take κ(u) = u and m = 1 in (13) In the following we give another version of the Hadamard inequality for generalized fractional integral operators.

Corollary 2.
Under the assumptions of Theorem 9 if we take m = 1, then we get following inequalities for exponentially convex function: Remark 3.

Fejér-Hadamard Inequalities for exponentially m-convex functions
Here we give two versions of the Fejér-Hadamard inequality.

Corollary 3.
Under the assumptions of Theorem 10 if we take m = 1, then we get following inequalities for exponentially convex function:

Remark 4.
1. If we take κ(u) = u in (25), then Theorem 7 is obtained. 2. If we take κ(u) = u and m = 1 in (25) In the following we give another generalized fractional version of the Fejér-Hadamard inequality.