On Hermite-Hadamard Type Inequalities for s-Convex Functions on the Coordinates via Riemann-Liouville Fractional Integrals

We obtain some Hermite-Hadamard type inequalities for -convex functions on the coordinates via Riemann-Liouville integrals. Some integral inequalities with the right-hand side of the fractional Hermite-Hadamard type inequality are also established.


Introduction
If  :  →  is a convex function on the interval , then, for any ,  ∈  with  ̸ = , we have the following double inequality: This remarkable result is well known in the literature as the Hermite-Hadamard inequality.Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1][2][3][4][5][6][7]).
It can be easily seen that, for  = 1, -convexity reduces to ordinary convexity of functions defined on [0, ∞).
In [4], Dragomir defined convex functions on the coordinates as follows.
A formal definition for convex functions on the coordinates may be stated as follows.
A formal definition for convex functions on the coordinates in the second sense may be stated as follows.Definition 4. A function  : Δ → R is said to be -convex on coordinates in the second sense on Δ if the inequality, holds for all (, ), (, ), (, ), (, ) ∈ Δ with ,  ∈ [0, 1] and some fixed  ∈ (0, 1].
In [10], Alomari and Darus proved the following inequalities based on the above definition.
Theorem 5 (see [10]).Suppose that the coordinates in the second sense on Δ.Then one has the inequalities: It is remarkable that Sarikaya et al. [11] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.
We remark that the symbol    + and    −  denote the leftsided and right-sided Riemann-Liouville fractional integrals of the order  ≥ 0 with  ≥ 0 which are defined by respectively.Here, Γ() is the Gamma function defined by Definition 7 (see [12]).Let  ∈ In [12], Sarıkaya proposed the following Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals by using convex functions of two variables on the coordinates.Theorem 8 (see [12]).Let  : Δ ⊆ R 2 → R be convex functions on the coordinates on Δ = [, ] × [, ] in R 2 with 0 ≤  < , 0 ≤  < , and  ∈  1 (Δ).Then one has the inequalities: In [12], Sarıkaya established some Hermite-Hadamard inequalities for convex functions on the coordinates in the second sense via fractional integrals based on the following lemma.
Lemma 9 (see [12]). where In this paper, we establish some Hermite-Hadamard type inequalities for -convex functions on the coordinates functions via Riemann-Liouville integrals.Some integral inequalities with the right-hand side of the fractional Hermite-Hadamard type inequality are also given.

Fractional Inequalities for 𝑠-Convex
Functions on the Coordinates Proof.From (7), with then integrating the resulting inequality with respect to  1 ,  2 over [0, 1] × [0, 1], we obtain Using the change of the variable, we get by which the first inequality is proved.For the proof of the second inequality, we note that  is -convex on coordinates, then Multiplying both sides of above inequalities by , then integrating the resulting inequality with respect to The proof is completed.
Remark 11.Applying Theorem 10 for  = 1, we get Theorem 8. We note that the Beta functions is defined by

Inequalities for Differentiable Functions
which completes the proof.
We get the desired results.

Conclusion
In this paper, we obtain some Hermite-Hadamard type inequalities for coordinated -convex functions via Riemann-Liouville integrals.An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for other kinds of convex functions on the coordinates via Riemann-Liouville integrals.