Hermite-Hadamard and Simpson's type of inequalities for first and second ordered derivatives using product of (h₁, h₂, s)-convex and m-convex function

This article is dedicated to find the extensions for Hermite-Hadamard (H-H) and Simpson's type of inequalities. By combining multiple existing convex functions by placing specific restrictions on them is the most effective in many approaches to find a new convex function. Here, to find the new function (h(1), h(2), s)-Convex and m-Convex Function are used. Because of the product of two or even more convex functions does not necessarily have to be convex, we decided to investigate merging distinct convex functions. Combining more than two convex functions in a novel adaptive way advances to new applications in a range of disciplines, including mathematics and other fields. In this paper, some extensions for Hermite-Hadamard and Simpson's inequalities is explored. The newly constructed extensions of these inequalities will be considered as the improvements and refinements of previously obtained results.