On the Hermite-Hadamard-Fejer-type inequalities for co-ordinated convex functions via fractional integrals

Article History: Received 20 January 2017 Accepted 06 June 2017 Available 17 July 2017 In this paper, using Riemann-Liouville integral operators, we establish new fractional integral inequalities of Hermite-Hadamard-Fejer type for coordinated convex functions on a rectangle of R. The results presented here would provide extensions of those given in earlier works.


Introduction
Let Φ : I ⊆ R → R be a convex mapping defined on the interval I of real numbers and a, b ∈ I, with a < b.The following double inequality is well known in the literature as the Hermite-Hadamard inequality [13]: The most well-known inequalities related to the integral mean of a convex function are the Hermite Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fejér inequalities (see, [14], [19], [21]).In [11], Fejer gave a weighted generalizatinon of the inequalities (1) as the following: where Ψ : [a, b] → R is nonnegative, integrable and symmetric to (a+b) 2 .In the following, we will give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used further in this paper.More details, one can consult [12,18].Definition 1. ( [4,12,18]) Let Φ ∈ L 1 ([a, b]).The Riemann-Liouville integrals J α a+ Φ and J α b− Φ of order α > 0 with a ≥ 0 are defined by *Corresponding Author Later, in [14], Iscan presented the following Hermite-Hadamard-Fejer type inequalities for convex functions via Riemann-Liouville fractional integrals: integrable and symmetric with respect to a+b 2 , then the following inequalities for fractional integrals hold: with α > 0.
Let us now consider a bi-demensional interval which will be used throughout this paper.So, we define ∆ =: [a, b] × [c, d] in R 2 with a < b and c < d.A mapping Φ : ∆ → R is said to be convex on the co-ordinates ∆ if the following inequality: holds, for all (x, y) , (z, r) ∈ ∆ and t ∈ [0, 1] .
A function Φ : ∆ → R is said to be convex on the co-ordinates on ∆ if the partial mappings Φ y : ) are convex where defined for all x ∈ [a, b] and y ∈ [c, d] (see, [10]).
A formal definition for co-ordinated convex functions may be stated as follows: Definition 2. ( [10]) A function Φ : ∆ → R will be called co-ordinated convex on ∆, for all t, s ∈ [0, 1] and (x, y), (u, r) ∈ ∆, if the following inequality holds: +t Clearly, every convex function is a co-ordinated convex.Furthermore, there exists a co-ordinated convex function which is not convex, (see, [10]).
In [10], Dragomir established the following inequality of Hermite-Hadamard-type for coordinated convex mapping on a rectangle of R 2 similar to (1).
Theorem 4. Suppose that Φ : ∆ → R is coordinated convex on ∆.Then one has the inequalities: The above inequalities are sharp.
Later, in [27], Sarikaya and Yaldiz proved inequalities of the Hermite-Hadamard type by using the definition of co-ordinated convex functions for L-Lipschitzian mappings.
In [3], a Hermite-Hadamard-Fejer type inequality for co-ordinated convex mappings is established as follows: Theorem 5. Let Φ : ∆ → R be a co-ordinated convex function.Then the following double inequality hold: where p : ∆ → R is positive, integrable and symmetric with respect to x = a+b 2 and y = c+d 2 on the co-ordinates on ∆.The above inequalities are sharp.
Because of the wide application of Hermite Hadamard type inequalities, Fejer type inequalities and Riemann-Liouville integrals for twovariable functions, many authors extend their studies to Hermite Hadamard type inequalities and Fejer type inequalities involving Riemann-Liouville integrals not limited to integer integrals.
x < b, y < d respectively.Here, Γ is the Gamma function, and Similar to Definition 1 and Definition 3, we introduce the following fractional integrals: It is remarkable that Sarikaya et al.( [26]) and ( [28]) gave the following interesting integral inequalities of Hermite-Hadamard-type involving Riemann-Liouville fractional integrals by using convex functions of 2-variables on the coordinates.
The main aim of this paper is to establish new results on Hermite-Hadamard-Fejer type inequalities for co-ordinated convex functions on the rectangle ∆ introduced in the first section of this paper.We will use the Riemann-Liouville integral operators to prove our main results.

Hermite-Hadamard-Fejer type inequalities for fractional integrals
In this section, using Riemann-Liouville fractional integral operators, we establish new results on Hermite-Hadamard-Fejer type inequalities for coordinated convex functions.We present evidence by using two different methods.We begin by the following theorem: R is nonnegative, integrable and symmetric with respect to a+b 2 , c+d 2 on the co-ordinates, then for any α, β > 0 with a, c ≥ 0, the following integral inequalities hold Proof.Since Φ is a convex function on ∆, then, for all (t, s) ∈ [0, 1] × [0, 1] , we can write: Multiplying both sides of (9) by , and integrating the resulting inequality with respect to Therefore, The first inequality of ( 8) is thus proved.We shall prove the second inequality of (8): Since f is a convex function on ∆, then, for all (t, s) ∈ ≤ Φ(a, c) + Φ(b, c) + Φ(a, d) + Φ(b, d).
We prove also the following result: negative, integrable and symmetric with respect to a+b 2 and c+d 2 on the co-ordinates, then we have: Proof.Since Φ : ∆ → R is convex on the co-ordinates, it follows that the mapping y) is nonnegative, integrable and symmetric with respect to c+d 2 , for all x ∈ [a, b].Then, thanks to the inequalities (3), we can write Multiplying both sides of ( 12) by and , and integrating with respect to x over [a, b], respectively, we have Adding the inequalities ( 13)-( 17), we can write These give the second and the third inequalities in (11).Now, by using the first inequality in (3), it yields that which gives the first inequality in (11).
Finally, by using the second inequality in (3), we can state that: By addition, we get the last inequality in (11).

Conclusion
In this paper, we established the Hermite-Hadamard-Fejer type inequalities for coordinated mappings related results to present new type of inequalities involving Riemann-Liouville integral operator.The results presented in this paper would provide generalizations of those given in earlier works.The findings of this study have several significant implications for future applications.

Theorem 1 .
Let Φ : [a, b] → R be a convex function.Then the inequality hold: