Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on coordinated convex functions in this work. In this paper, we first introduce Simpson’s second-type integral inequalities for two-variable functions whose second-order partial derivatives in modulus are convex on the coordinates. In addition, similar results are acquired by considering that powers of the absolute value of second-order partial derivatives of these two-variable functions are convex on the coordinates. Finally, some applications for Simpson’s $3/8$ cubature formula are given.

This study seeks to derive novel quantum variations of Simpson’s inequality by primarily utilizing the convexity characteristics of functions. Additionally, the study examines the credibility of the obtained results through the presentation of relevant numerical examples and graphs.