The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the well-known fuzzy Hermite–Hadamard (HH) type and associated inequalities. With the help of fuzzy Aumann integrals and the newly introduced fuzzy number valued up and down convexity ($U\mathcal{D}$-convexity), we increase this mileage even further. Additionally, with the help of definitions of lower $U\mathcal{D}$-concave (lower $U\mathcal{D}$-concave) and upper $U\mathcal{D}$-convex (concave) fuzzy number valued mappings ($\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s), we have gathered a sizable collection of both well-known and new extraordinary cases that act as applications of the main conclusions. We also offer a few examples of fuzzy number valued $U\mathcal{D}$-convexity to further demonstrate the validity of the fuzzy inclusion relations presented in this study.

Convex bodies are naturally symmetrical. There is also a correlation between the two variables of symmetry and convexity. Their use, in either case, has been feasible in recent years because of their interchangeable and similar properties. The proposed analysis provides information on a new class for a convex function which is known as up and down $\left({\mathcal{X}}_1,{\mathcal{X}}_2\right)$-convex fuzzy-Number valued mappings ($UD$-$\left({\mathcal{X}}_1,{\mathcal{X}}_2\right)$-convex $FNVM$). Using this class, we disclosed a number of new versions of integral inequalities. Additionally, we give a number of new related integral inequalities connected to the well-known Hermite-Hadamard-type inequalities. In conclusion, some examples are given to back up and show the value of these new results.