New conformable fractional integral inequalities of Hermite-Hadamard type for convex functions

In this work, we established new inequalities of Hermite-Hadamard type for convex functions via conformable fractional integrals. Through the conformable fractional integral inequalities, we found out some new inequalities of Hermite-Hadamard type for convex functions in a form of classical integrals.


Introduction
A function h : I ⊆ R → R is said to be convex on the interval I, if the inequality holds for all x, y ∈ I and ζ ∈ [0, 1].We say that h is concave if −h is convex.For convex functions, many equalities or inequalities have been established by many authors; for example Ostrowski type inequality [19], Hardy type inequality [20], Olsen type inequality [21] and Gagliardo-Nirenberg type inequality [22] but the most common and significant inequality is the Hermite-Hadamard type inequality [12,17], which is defined as: where the function h : I ⊆ R → R is assumed to be a convex function with u < v and u, v ∈ I.
A number of mathematicians in the field of applied and pure mathematics have dedicated their efforts to extend, generalize, counterpart and refine the Hermite-Hadamard inequality (1.2) for different classes of convex functions and mappings.For more recent results obtained on inequality (1.2); we refer the reader to [10][11][12][13][14]. Definition 1. [8] Suppose that h ∈ L ([u, v]).The left and right Riemann-Liouville fractional integrals J α u + h and J α v − h of order α > 0 are defined by and respectively, where Γ(α) is the standard gamma function defined by Γ(α) = ∞ 0 e −ζ ζ α−1 dζ and J 0 u + h(x) = J 0 v − h(x) = h(x).
As for classical integrals, many Hermite-Hadamard type inequalities have been established for the Riemann-Liouville fractional integrals; for more details and interesting applications see [15][16][17].Now, we give definition of conformable fractional derivative with its important properties which are useful in order to obtain our main results (see [1][2][3][4][5][6][7]18]). In our study, we use the Katugampola derivative formulation of conformable derivative which is explained in the following definition: Definition 2 ([7]).Given a function h : [0, ∞) → R. Then the conformable fractional derivative of h of order α of h at ζ is defined by Also, note that if h is differentiable, then We can write to denote the conformable fractional derivatives of h of order α at ζ.In addition, if the conformable fractional derivative of h of order α exists, then we simply say h is α-differentiable.Theorem 3 ([7]).Let α ∈ (0, 1] and h, g be α-differentiable at a point ζ > 0. Then Now, we give the definition of conformable fractional integral: exists and is finite.
). (v) For the usual Riemann improper integral and α ∈ (0, 1], we have The aim of our article is to establish some new inequalities connected with the Hermite-Hadamard inequalities (1.2) via conformable fractional integral.

Main Results
Our main results depend on the following equality: ), then the following identity for conformable fractional integral holds: where and Proof.By using definition of conformable fractional derivative (1.4), we have On integrating by parts one can have Using the change of the variable
Remark 7.With the similar assumptions of Lemma 6, if α = 1, then identity (2.1) reduces to the following identity: where which is obtained by Shi et al. [15].
then the following inequality for conformable fractional integral holds: Proof.Using Lemma 6 and the property (1.4), we have By using the convexity of x α−1 for x > 0, α ∈ (0, 1], we have and This completes the proof of Theorem 8.

Corollary 9.
With the similar assumptions of Theorem 8, if α = 1, then then the following inequality for conformable fractional integral holds: Proof.Using Lemma 6 and the property (1.4), we have It follows from the power-mean inequality that where we have used the facts that and Analogously .

Preprints
then the following inequality for conformable fractional integral holds:

Conclusion
In this work, we have established new conformable fractional integral inequalities of Hermite-Hadamard type for convex functions using the.As a special case, if we substitute α = 1 into the general definition of conformable fractional integrals (Definition 2), we obtain the classical integrals.In view of this, we obtained some new inequalities of Hermite-Hadamard type for convex functions involving classical integrals.