New Estimates for Exponentially Convex Functions via Conformable Fractional Operator

In this paper, we derive a new Hermite–Hadamard inequality for exponentially convex functions via α-fractional integral. We also prove a new integral identity. Using this identity, we establish several Hermite–Hadamard type inequalities for exponentially convexity, which can be obtained from our results. Some special cases are also discussed.


Introduction
Convexity theory has played fundamental parts in the developments of various fields of pure and applied sciences.Due to its impermanence, convex functions and convex sets have been generalized and extended in different directions.It has been shown that a function is convex, if and only if, it satisfies an integral inequality, which his called the Hermite-Hadamard inequality.On other hand, the minimum of the differentiable convex functions can be characterized by variational inequalities.These two aspects of the convexity theory have far reaching applications and have provided powerful tools for studying difficult problems.In recent years, integral inequalities are being derived via fractional analysis, which has emerged as another interesting technique.Fractional analysis is an area that is constantly developing and trying to renew itself to produce solutions to the changing world and problems.Various types of fractional derivative and integral operators were studied.In fractional calculus, the fractional derivatives are defined via fractional integrals.The conformable fractional integral plays a major role in fractional calculus.There were several studies in the literature that include further properties such as expansion formulas, variational calculus applications, control theoretical applications, convexity and integral inequalities and Hermite-Hadamard type inequalities of this new operator and similar operators.
Exponentially convex functions have emerged an a significant new class of convex functions, which have important applications in technology, data science and statistics.The main motivation of this paper depends on a new identity that has been proved via α-fractional integrals (conformable fractional integral operators) and applied for exponentially convex functions.This identity offers new upper bounds and estimations of Hadamard type integral inequalities.Some special cases such for α = 1 have been discussed, which can be deduced from these results.In derivation of these results, we have used integration techniques, some integral inequalities as power-mean inequality and Jensen inequality.
We now recall some well known concepts and basic results, which are needed in the derivation of our results.
In recent years, the concept of convexity has been extended and generalized in various directions using novel and innovative ideas and techniques to study a a wide class of unrelated problems in a unified framework.Awan et al. [5] considered and studied a new class of exponentially convex functions.Antczak [13] explored the applications of the exponentially convex functions in the mathematical programming problems.Dragomir and Gomm [7] derived some integral inequalities for the exponentially convex functions.
We now recall the definition of exponentially convex function.
One can easily show that the minimum u ∈ K is the minimum of the differentiable exponentially convex functions f , if and only if, u ∈ K satisfies the inequality which is called the exponentially variational inequality and appears to be new one.
It is an open problem to study the exponentially variational inequalities and their properties.For the applications, numerical methods and other aspects of variational inequalities, see Noor [16].
An important definition called Riemann-Liouville fractional integrals which is a milestone in the theory of fractional calculus: In the case of α = 1, the fractional integral reduces to classical integral.Several researchers have focused on new integral inequalities involving Riemann-Liouville fractional integrals in recent years, see the papers [3,10,[17][18][19][20][21][22][23].Recently Khalil et al. [22] gave a new definition that is called the "conformable fractional derivative" and its properties.The conformable fractional derivative attracts attention with conformity to the classical derivative.Khalil et al. [22] have introduced the conformable fractional derivative by the equation which has a limit form similar to the classical derivative.Khalil et al. [22] have proved that this definition provides multiplication and division rules.They also express the Roll's theorem and the mean value theorem for functions which are differentiable with conformable fractional order.Now, we give the definition of the conformable fractional derivative with its important properties which are useful in order to obtain our main results see, [18][19][20]22].
In our study, we use the Katugampola derivative formulation of conformable derivative, which is explained in the following definition: Definition 5. ( [21]) Given a function f : [0, ∞) → R.Then, the conformable fractional derivative of f of order α of f at t is defined by If f is α-differentiable in some (0, α), α > 0, lim t→0 + h (α) (t) exist, then define Note that, if f is differentiable, then We can write f α (t) for D α ( f )(t) denotes the conformable fractional derivatives of f of order α at t.If the conformable fractional derivative of f of order α exists, then we simply say f is α-differentiable.
Khalil et al. [22] considered the following definition: exists and is finite.
The motivation of this article is to discuss some new fractional bounds involving the functions having exponential convexity property.In order to obtain main results of the article, we derive several new conformable fractional integral identities.We hope that the ideas and techniques of this article will inspire interested readers.

Results
Our main results depend on the following inequality: where and Proof.Using integration by parts, we have Using the change of the variable x := at + (1 − t) 3a+b 4 , t ∈ [0, 1] and the definition of conformable fractional integral (4), we obtain Similarly, we get and Adding θ 1 , θ 2 θ 3 and θ 4 together, we obtain the desired inequality (5).
Remark 2. If we set α = 1, then under the assumption of Lemma 1, the identity (5) reduces to the the following new identity. where where and Proof.Using Lemma 1 and the convexity of | f |, we find By using the convexity of x α−1 for x > 0, α ∈ (0, 1], we have Using (8), ( 9), ( 10) and ( 11) in (7) and the properties of modulus, we get Thus (12) and ( 13) become Simple calculations yield which is the required result. and Proof.Using Lemma 1 and the concavity of | f | q , we find It follows from the power-mean inequality that Since | f | q is exponentially concave on [a, b] for any t ∈ [0, 1], we obtain , where we have used the fact that and Analogously: .

Conclusions
In this paper, we have established several new conformable fractional integral inequalities of Hermite-Hadamard type for exponentially convex functions.If α = 1, then, one can obtain the classical integrals (as a special case) from the general definition of Conformable fractional integrals.Consequently, we have obtained some new inequalities of Hermite-Hadamard type for exponentially convex functions involving classical integrals.The ideas and techniques of this paper may stimulate further research in this dynamic field.

Theorem 3 .
Let f : [a, b] ⊆ R → R be an α-fractional exponentially differentiable mapping on (a, b) with 0