Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals


 In this study, we first obtain a new identity for generalized fractional integrals which contains some parameters. Then by this equality, we establish some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals. Moreover, we show that the results proved in the main section reduce to several Simpson-, trapezoid- and midpoint-type inequalities for various values of parameters.


Introduction
The inequalities discovered by C. Hermite and J. Hadamard for convex functions are considered significant in the literature. These inequalities state that if I : → is a convex function on the interval I of real numbers and κ κ I , 1 2 ∈ with κ κ 1 2 < , then  Both inequalities hold in the reversed direction if is concave. Over the last 20 years, numerous studies have focused on obtaining trapezoid-and midpoint-type inequalities which give bounds for the right-hand side and left-hand side of the inequality (1.1), respectively. For example, Dragomir and Agarwal first obtained trapezoid inequalities for convex functions in [1], whereas Kirmacı first, established midpoint inequalities for convex functions in [2]. In [3], Sarikaya et al. generalized the inequalities (1.1) for fractional integrals and the authors also proved some corresponding trapezoid-type inequalities. Iqbal et al. presented some fractional midpoint-type inequalities for convex functions in [4]. On the other hand, Dragomir proved Hermite-Hadamard inequalities for co-ordinated convex mappings in [5]. In [6] and [7], the authors proved midpoint-and trapezoid-type inequalities for co-ordinated convex functions, respectively. Moreover, Sarikaya obtained fractional Hermite-Hadamard inequalities and fractional trapezoid for functions with two variables in [8]. Tunç et al. presented some fractional midpoint-type inequalities for co-ordinated convex functions in [9]. In [10], Sarikaya and Ertuğral first introduced new fractional integrals which are called generalized fractional integrals, and then, they proved Hermite-Hadamard inequalities and several trapezoid-and midpoint-type inequalities for generalized fractional integrals. In addition, Turkay et al. defined the generalized fractional integrals for functions with two variables and they presented Hermite-Hadamard-and trapezoid-type inequalities for this kind of fractional integrals in [11]. For the other similar inequalities, please refer to [12][13][14][15][16][17][18][19].
On the other hand, the following inequality is well known in the literature as Simpson's inequality.
The aim of this paper is to obtain some parameterized inequalities for co-ordinated convex functions via generalized fractional integrals. These inequalities reduce to Simpson, trapezoid and midpoint inequalities in the case of special choice of parameters. The overall structure of the study takes the form of six sections including an introduction. The remaining part of the paper proceeds as follows: In Section 2, the generalized fractional integral operators is summarized, along with some related theorems. In Section 3, an identity involving generalized fractional integrals is presented for partial differentiable functions. Then we prove several parameterized inequalities for functions whose partial derivatives in absolute value are co-ordinated convex in Section 4. Moreover, some special cases of the results in Section 4 are presented in Section 5. Finally, some conclusions and further directions of research are discussed in Section 6.
A formal definition for co-ordinated convex function may be stated as follows: The mapping is a co-ordinated concave on Δ if the inequality (1.2) holds in the reversed direction for all t s , 0,1 ∈ [ ] and x u y v , , , Δ ∈ ( ) ( ) .

Generalized fractional integrals
Fractional calculus and applications have application areas in many different fields such as physics, chemistry and engineering as well as mathematics. The application of arithmetic carried out in classical analysis in fractional analysis is very important in terms of obtaining more realistic results in the solution of many problems. Many real dynamical systems are better characterized by using non-integer order dynamic models based on fractional computation. While integer orders are a model that is not suitable for nature in classical analysis, fractional computation in which arbitrary orders are examined enables us to obtain more realistic approaches.
In this section, we summarize the generalized fractional integrals defined by Sarikaya and Ertuğral in [10].
Definition 2. Let κ κ : , 1 2 → [ ] be an integrable function. The left-sided and right-sided generalized fractional integral operators are given by respectively. Here, the function φ : 0, 0, The most important feature of generalized fractional integrals is that they generalize some types of fractional integrals such as Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral, Katugampola fractional integrals, conformable fractional integral, Hadamard fractional integrals, etc. These important special cases of the integral operators (2.1) and (2.2) are mentioned below.
In this definition, known fractional integrals can be obtained by some special choices. For example, (i) If we take φ ξ ξ = ( ) and ψ η η = ( ) , then the operators (2.3), (2.4), (2.5) and (2.6) transform into the Riemann integrals on coordinates, respectively, as follows: where Γ is the gamma function. transform into the Riemann-Liouville k-fractional integrals on coordinates [48], respectively, as follows: where Γ k is the k-gamma function.

An Identity
Throughout this study, for brevity, we define Now we give an identity for generalized fractional integrals.
Similarly, we get By the equalities (3.2)-(3.5), we have in Lemma 1, we obtain, which is given by Budak and Ali in [49].
then we obtain the following new Riemann-Liouville fractional integral identity: then we obtain the following new k-Riemann-Liouville fractional integral identity: where κ κ κ κ , ; ,

Some new inequalities for generalized fractional integrals
In this section, we establish some new Simpson-type inequalities for differentiable co-ordinated convex functions via generalized fractional integrals.
Some new parameterized inequalities  1163 Theorem 2. We assume that the conditions of Lemma 1 hold. If the mapping ξ η 2 ∂ ∂ ∂ is co-ordinated convex on κ κ , 1 2 [ ], then the following inequality holds for generalized fractional integrals: Proof. By taking the modulus in Lemma 1 and using the properties of the modulus, we obtain that Since the mapping ξ η 2 ∂ ∂ ∂ is co-ordinated convex on Δ, therefore, we have Similarly, we obtain By the inequalities (4.4)-(4.7), the proof is completed.
then we obtain the following parameterized Simpson-type inequality for k-Riemann-Liouville fractional integrals: where κ κ κ κ , ; , I( )is defined as in Corollary 2.
Theorem 5. We assume that the conditions of Lemma 1 hold. If the mapping ξ η is co-ordinated convex on Δ, then we have the following inequality: Proof. Taking the modulus in Lemma 1 and using the Hölder inequality, where the mapping w : 0, 1 0, 1

Special cases of main results
In this section, we present some special cases of the results proved in the previous section.  By special choices of λ 1 , λ 2 , μ 1 and μ 2 in Theorems 3, 4 and 5, one can obtain several new Simpson-, trapezoid-and midpoint-type inequalities. Writing these situations is left to the reader as it will make the article too long.

Conclusion
In this paper, we present several generalized inequalities for co-ordinated convex functions via generalized fractional integrals. It is also shown that the results given here are the strong generalization of some already published ones. It is an interesting and new problem that the forthcoming researchers can use the techniques of this study and obtain similar inequalities for different kinds of co-ordinated convexity in their next works.
Acknowledgements: The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.
Funding information: The funding of this work was supported by Zhejiang Normal University, Jinhua, China.
Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.