Fractional Hadamard and Fejér-Hadamard Inequalities Associated with Exponentially ðs,mÞ-Convex Functions

School of Mathematics and Big Data, Chongqing University of Arts and Sciences, Chongqing 402160, China Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan Government Boys Primary School Sherani, Hazro, Attock, Pakistan Department of Mathematics, GC University Lahore, Pakistan


Introduction and Preliminaries
C onvex functions are very important in the field of mathematical inequalities. Nobody can deny the importance of convex functions. A large number of mathematical inequalities exist in literature due to convex functions. For more information related to convex functions and it's properties (see, [1][2][3]).

Definition 1.
A function µ : I → R on an interval of real line is said to be convex, if for all α, β ∈ I and κ ∈ [0, 1], the following inequality holds: The function µ is said to be concave if −µ is convex.
Next we give the definition of exponentially convex functions.
Definition 4. [9,22] A function µ : I → R on an interval of real line is said to be exponentially convex, if for all α, β ∈ I and κ ∈ [0, 1], the following inequality holds: In [23], Rashid et al., gave the definition of exponentially s-convex functions.
Definition 5. Let s ∈ [0, 1]. A function µ : I → R on an interval of real line is said to be exponentially s-convex, if for all α, β ∈ I and κ ∈ [0, 1], the following inequality holds: In [24], Rashid et al., gave the definition of exponentially h-convex functions.

Definition 6.
Let J ⊆ R be an interval containing (0, 1) and let h : J → R be a non-negative function. Then a function µ : I → R on an interval of real line is said to be exponentially h-convex, if for all α, β ∈ I and κ ∈ [0, 1], the following inequality holds: In [25], Rashid et al., gave the definition of exponentially m-convex functions.
In [33], Mehmood et al., proved the following formulas for constant function: The objective of this paper is to establish the Hadamard and the Fejér-Hadamard inequalities for generalized fractional integral operators (12) and (13) containing Mittag-Leffler function via a monotone function by using the exponentially (h, m)-convex functions. These inequalities lead to produce the Hadamard and the Fejér-Hadamard inequalities for various kinds of exponentially convexity and well known fractional integral operators given in Remark 1 and Remark 2. In Section 2, we prove the Hadamard inequalities for generalized fractional integral operators (12) and (13) via exponentially (h, m)-convex functions. In Section 3, we prove the Fejér-Hadamard inequalities for these generalized fractional integral operators via exponentially (h, m)-convex functions. Moreover, some of the results published in [26,33,34] have been obtained in particular.

Fractional Hadamard inequalities for exponentially (h, m)-convex functions
In this section, we will give two versions of the generalized fractional Hadamard inequality. To establish these inequalities exponentially (h, m)-convexity and generalized fractional integrals operators have been used.

Corollary 2.
Setting m = 1 in (22), the following inequalities for exponentially h-convex function can be obtained: whereψ is same as in (21).

Fractional Fejér-Hadamard Inequalities for exponentially (h, m)-convex functions
In this section, we will give two versions of the generalized fractional Fejér-Hadamard inequality. To establish these inequalities exponentially (h, m)-convexity and generalized fractional integrals operators have been used.

Corollary 3.
Setting m = 1 in (28), the following inequalities for exponentially h-convex function can be obtained: whereψ is same as in (21). In the following we give another generalized fractional version of the Fejér-Hadamard inequality.

Concluding remarks
In this article, we established the Hadamard and the Fejér-Hadamard inequalities. To established these inequalities generalized fractional integral operators and exponentially (h, m)-convexity have been used. The presented results hold for various kind of exponentially convexity and well known fractional integral operators given in Remarks 1 and 2. Moreover, the established results have connection with already published results.