Some Estimates for Generalized Riemann-Liouville Fractional Integrals of Exponentially Convex Functions and Their Applications

: In the present paper, we investigate some Hermite-Hadamard ( HH ) inequalities related to generalized Riemann-Liouville fractional integral ( GRLFI ) via exponentially convex functions. We also show the fundamental identity for GRLFI having the ﬁrst order derivative of a given exponentially convex function. Monotonicity and exponentially convexity of functions are used with some traditional and forthright inequalities. In the application part, we give examples and new inequalities for the special means.


Introduction
Recently, several researchers have attracted the fractional calculus, see References [1][2][3][4]. The effect and motivation of this fractional calculus in both theoretical and applied science and engineering rose substantially. Numerous studies related to the discrete versions of this fractional calculus have been established, which benefit from countless applications in the theory of time scales, physics, different fields of engineering, chemistry and so forth (e.g, see References  and the references therein).
A few decades ago, a lot of new operator definitions were given and the properties and structures of these operators have been examined. Some of these operators are very close to classical operators in terms of their characteristics and definitions. It is known that the GRLF I, which was introduced in reference [33], extends several well-known fractional integral operators (see Remark 1 below). Both the generalized Riemann-Liouville fractional derivative and the integral operator are useful in the study of transform theory, quantum theory and fractional intgerodifferential equations.
Almost every mathematician knows the importance of convexity theory in every field of mathematics, for example in nonlinear programming and optimization theory. By using the concept of convexity, several integral inequalities have been introduced such as Jensen, HH and Slater inequalities, and so forth. But the well-known one is the celebrated HH inequality.
Let I ⊆ R be an interval and U : I → R be a convex function. Then the double inequality and respectively; with Γ(.), the classical gamma function.

Remark 1.
Many known defined fractional integral operators are just special cases of (2) and (3).
The principal objective of this paper is to use a new convex class and a new integral operator to obtain new versions of HH-inequality that give bounds for the mean value of convexity. Also, we will establish some more general estimates and related modulus inequalities for GRLF I via exponentially convex functions. In addition, the accuracy of the results was tested with applications of special means and with some examples.

HH Inequality for GRLF I
is an increasing and positive monotone function on (d 1 , d 2 ], with continuous derivative Ψ (x) on (d 1 , d 2 ). Let U is an exponentially convex function and ρ ∈ (0, 1). Then Multiplying by τ ρ−1 on both sides of inequality (6) and then integrating w.
That is, Hence the proof is completed.
Our next result is the subsequent lemma, which is useful for our coming results.
is an increasing and positive monotone function on (d 1 , and ρ ∈ (0, 1). Then Proof. Consider By making a change of variable in the above equation Now It follows that This completes the proof.
be an increasing and positive monotone function on (d 1 , and ρ ∈ (0, 1). If |U | q is exponentially convex and q ≥ 1, then Proof. First note that, for every Applying Lemma 1, Hölder's inequality and exponentially convexity of |U |, we obtain where By substituting the above integral values in (12) and after some simplification, we get the required inequality (13).

Corollary 1.
Letting q = 1, then under the assumption of Theorem 2, we have Proof. Since Ψ is differentiable and strictly increasing function, we can write and τ ∈ [d 1 , x], ρ ≥ 1, and Ψ (τ) > 0. Then, the subsequent inequality holds true By exponentially convexity of U , we have From (14) and (15), one has By using (2) of Definition 2, we get and δ ≥ 1, the subsequent inequality holds true By exponentially convexity of U , we have Adopting the same procedure as we did for (14) and (15) , one can get from (17) and (18) the coming inequality From inequalities (16) and (19), we get (13). Hence the proof is completed.
Particular cases are stated as follows.

Corollary 2.
Choosing ρ = δ in (13), then we have a new inequality for GRLF I; Corollary 3. Choosing x = d 1 and x = d 2 in (13), adding the resulting inequalities, then the conditions of Theorem 1 are satisfied, we have

Corollary 4.
If we take ρ = δ in (20), then we get the following inequality for GRLF I −→ R be the functions such that e U be differentiable function, Ψ is also differentiable and strictly increasing with respectively.
Proof. From the convexity of |(e U ) |, we obtain Since Ψ is a differentiable and strictly increasing function, we have the subsequent inequality where as x ∈ [d 1 , d 2 ] and τ ∈ [d 1 , x], ρ > 0. From (25) and (26), one has Therefore (28) takes the form Also from (24), one has Following the same procedure as we did for (25), we also have From (29) and (31), we get By convexity of |(e U ) |, we have Now for x ∈ [d 1 , d 2 ] and τ ∈ [x, d 2 ] and δ > 0, the following inequality holds true Following the same way as we have done for (25), (26) and (30) we can get from (33) and (34) the subsequent inequality From inequalities (32) and (35) using triangular inequality, we get (21) which is desired.
Particular cases are stated as follows.

Corollary 5.
Choosing ρ = δ in (21), then we have a new inequality for GRLF I To prove our next result we need the following Lemma.
Proof. Write Since e U is convex, therefore we have Also, e U is symmetric about d 1 +d 2 2 , therefore we have e U (x) = e U (d 1 +d 2 −x) and the inequality in (36) holds.

Theorem 4.
Suppose that U : [d 1 , d 2 ] −→ R be an exponentially convex function such that e U is positive convex and symmetric about d 1 +d 2 2 , Ψ is a differentiable and strictly increasing function having Ψ ∈ L 1 ([d 1 , d 2 ]). Then ρ, δ > 0, we have Proof. Since Ψ is differentiable and strictly increasing function therefore , δ > 0, and Ψ (x) > 0. Hence, the following inequality holds true From the exponential convexity of U , it can be obtained From (38) and (39), one can have By using (2) of Definition 2, we get and ρ > 0, the following inequality holds true Adopting the same procedure as we did for (38) and (39) one can get from (40) and (42) the subsequent inequality From (41) and (43), we get Using Lemma 3 and multiplying (36) we get By using (2) of Definition 2 we get Similarly, using Lemma 3 and multiplying (36) From (44) and (47), we get (37) which is the required result.

Corollary 6.
Choosing ρ = δ in (37), then we have a new inequality for GRLF I

Conclusions
In this article, we have investigated a few fractional integral inequalities for GRLF I via exponentially convexity. These inequalities have bounds of the sum of left-sided and right-sided fractional integrals and inequalities for the function, and their first derivative in absolute value is exponentially convex. Also, fractional inequalities of HH type for a symmetric and exponentially convex function are proved. These estimates, bounds and inequalities exist for all fractional operators are stated in Remark 1. The method followed to produce fractional inequalities is innovative and simple. It could be followed to broaden further consequences for other classes of functions related to exponentially convex functions, using convenient fractional integral operators.
Author Contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.