In this work, we introduce the concept of double quantum integrals for the interval-valued functions of two variables. We offer several new inclusions of the Hermite-Hadamard type for co-ordinated convex interval-valued functions using the newly defined integrals. Moreover, we prove trapezoidal type inequalities for interval-valued functions of two variables using the ideas of the Pompeiu--Hausdorff distance between the intervals. It is also revealed that the results offered in this work are the generalization of several existing results.

In this article, we introduce a new concept of quantum integrals which is called $^{\kappa_{2}}T_{q}$-integral. Then we prove several properties of this concept of quantum integrals. Moreover, we present several Hermite-Hadamard type inequalities for $^{\kappa_{2}}T_{q}$-integral by utilizing differentiable convex functions. The results presented in this article are unification and generalization of the comparable results in the literature.

In this research paper, we investigate generalized fractional integrals to obtain midpoint type inequalities for the co-ordinated convex functions. First of all, we establish an identity for twice partially differentiable mappings. By utilizing this equality, some midpoint type inequalities via generalized fractional integrals are proved. We also show that the main results reduce some midpoint inequalities given in earlier works for Riemann integrals and Riemann-Liouville fractional integrals. Finally, some new inequalities for $k$-Riemann-Liouville fractional integrals are presented as special cases of our results.

In this paper, we introduce the notion of generalized fractional integrals for the interval-valued functions of two variables. We establish Hermite-Hadamard type inequalities and some related inequalities for co-ordinated convex interval-valued functions by using the newly defined integrals. It is also proved that the results given in this paper are the strong generalization of already published ones.

We first get some new Fejér type inclusions for products of interval-valued convex mappings. The most important feature of our work is that it contains Fejér type inclusions for both interval-valued integrals and interval-valued fractional integrals.