LAUSR

Numerous applications of the theory of convex and nonconvex mapping exist in the fields of applied mathematics and engineering. In this paper, we have defined a new class of nonconvex functions which is known as up and down pre-invex (pre-incave) fuzzy number valued mappings (F-N-V∙Ms). The well-known fuzzy Hermite–Hadamard (𝐻𝐻)-type and related inequalities are taken into account in this work. We extend this mileage further using fuzzy Riemann integrals and the fuzzy number up and down pre-invexity. Additionally, by imposing some light restrictions on pre-invex (pre-incave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down pre-invexity (pre-incavity), which are referred to as lower up and down pre-invex (pre-incave) and upper up and down pre-invex (pre-incave) fuzzy number valued mappings. By using these definitions, we have amassed a large number of both established and novel exceptional situations that serve as implementations of the key findings. To support the validity of the fuzzy inclusion relations put out in this research, we also provide a few examples of fuzzy numbers valued up and down pre-invexity.