On Bounds of fractional integral operators containing Mittag-Le ﬄ er functions for generalized exponentially convex functions

: Recently, a generalization of convex function called exponentially ( α, h − m )-convex function has been introduced. This generalization of convexity is used to obtain upper bounds of fractional integral operators involving Mittag-Le ﬄ er (ML) functions. Moreover, the upper bounds of left and right integrals lead to their boundedness and continuity. A modulus inequality is established for di ﬀ erentiable functions. The Hadamard type inequality is proved which shows upper and lower bounds of sum of left and right sided fractional integral operators.


Introduction
Convexity is one of the fascinating and natural concepts, it is beneficial in optimization theory, theory of inequalities, numerical analysis, economics and in other subjects of pure and applied mathematics. Convex functions are defined in different ways due to their interesting graphical shapes in euclidean space. A convex function defined on an interval of real line is always continuous in the interior points, but need not be differentiable. Although, it has left and right increasing derivatives at each interior point. The derivative of a differentiable convex function is always an increasing function. A twice differentiable convex function has downward concavity. In analytical forms it is defined in several ways the classical one is given in the following definition. Definition 1. A function φ : M ⊆ R → R, where M is convex set, is said to be convex function, if the following inequality holds: for all u, v ∈ M and t ∈ [0, 1].
The inequality (1.1) motivates the reader to extend, refine, generalize the notion of convexity. The authors have analyzed this inequality to introduce several new notions, for example m-convex function, s-convex function, h-convex function, p-convex function and many others. In [1], the notion of exponential convex function is introduced.
where M is an interval, is said to be exponentially convex function, if we have the following inequality: for all u, v ∈ M, t ∈ [0, 1] and σ ∈ R.
In [2], the notion of h-convex function is introduced as follows: Definition 3. Let h : N ⊃ [0, 1] → R be a non-negative function. A function φ : M → R is said to be h-convex function, if the following inequality holds: for all u, v ∈ M and t ∈ [0, 1], where M and N are intervals in R.
In [3], the following definition of (h − m)-convex function is given.
In [5], the definition of (s, m)-convex function is given.
Farid et al. in [6] unified the all above definitions in a single notion called (α, h−m)-convex function. Definition 7. Let N ⊆ R be an interval containing (0, 1) and let h : N → R be a non-negative function. We say that φ : A further generalization namely exponentially (α, h − m)-convex function is given in [7]. Definition 8. Let N ⊆ R be an interval containing (0, 1) and let h : N → R be a non-negative function. We say that φ : The above definition of exponentially (α, h − m)-convex function unifies the definitions of convex, exponentially convex, m-convex, exponentially m-convex, s-convex, exponentially s-convex, h-convex, exponentially h-convex, (h − m)-convex, exponentially (h − m)-convex, (s, m)-convex, exponentially (s, m)-convex, (α, m)-convex, exponentially (α, m)-convex functions in a single inequality. The aim of this paper is to study the extended generalized fractional integral operators involving Mittag-Leffler (ML) functions for exponentially (α, h − m)-convex function. By using definition of exponentially (α, h − m)-convex function, bounds of these fractional integral operators are obtained. The results will hold at the same time for all convex functions explained in above. The well-known Mittag-Leffler function E ξ (.) for one parameter is defined as follows [8]: where t, ξ ∈ C, (ξ) > 0 and Γ(.) is the gamma function. It is a natural extension of exponential, hyperbolic and trigonometric functions. This function and its extensions appear as solution of fractional integral equations and fractional differential equations. It was further explored by Wiman, Pollard, Humbert, Agarwal and Feller, see [9]. For its generalizations and extensions by various authors, we refer the reader to [9][10][11][12][13].
Next we give the definition of the generalized fractional integral operator containing the extended generalized Mittag-Leffler function (1.5).

Main results
be a real valued function. If f is positive and exponentially (α, h − m)-convex, m ∈ (0, 1], then there exist σ, τ ∈ R such that for ξ, η ≥ 1, the following fractional integral inequality holds: x) and ξ ≥ 1. Then the following inequality holds: After multiplying (2.2) and (2.3) and then integrating over By using the definition of left integral operators, we get Similarly, on the other hand for t ∈ (x, v] and η ≥ 1, the following inequality holds: Again by using definition of exponentially (α, h − m)-convexity of φ, for τ ∈ R we have (2.6) Multiplying (2.5) with (2.6) and then integrating over By using the definition of right integral operators, we get Sum of inequalities (2.4) and (2.7) gives the required inequality (2.1).
Some particular results are stated in the following corollaries.
Theorem 3. Let φ : [u, v] −→ R, u < mv, be a real valued function. If φ is differentiable and |φ | is exponentially (α, h−m)-convex, m ∈ (0, 1], then there exist σ, τ ∈ R such that for ξ, η ≥ 1, the following fractional integral inequality for generalized integral operators (1.7) and (1.8) holds: Proof. For x ∈ [u, v] and t ∈ [u, x), by using the definition of exponentially (α, h − m)-convexity of |φ |, for σ ∈ R we have (2.15) From(2.15), we can write (2.16) Multiplication of (2.2) and (2.16), gives the following: The left hand side of (2.18) is computed as follows: substituting x − t = r, using the derivative property (1.6) of Mittag-Leffler function, we have now for x − r = t in second term of the right hand side of the above equation and then using (1.7), we get Therefore (2.18) becomes: Again from (2.15) we can write Similarly as we did for (2.16), one can obtain:

Conclusions
In this research, we present the bounds of fractional integral operators containing Mittag-Leffler (ML) functions by using exponential (α, h − m)-convexity. Also we provide the generalization of various results already determined in [20][21][22][23][24][25][26]. The boundedness and continuity of several known integral operators defined in [11-13, 15, 16] are also mentioned. Also we have established upper and lower bounds in the form of the Hadamard like inequality. The reader can derive a plenty of fractional integral inequalities for various kinds of convex functions.