In this paper, we prove a new identity involving the second derivative of the function and Riemann-Liouville fractional integrals. The newly established identity is then used to establish some new Simpson's type inequalities for twice differentiable convex functions. Finally, we give applications of special functions using the newly proved inequalities.

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.