Some new concepts in fuzzy calculus for up and down λ-convex fuzzy-number valued mappings and related inequalities

In recent years, numerous scholars have investigated the relationship between symmetry and generalized convexity. Due to this close relationship, generalized convexity and symmetry have become new areas of study in the field of inequalities. With the help of fuzzy up and down relation, the class of up and down $\lambda$-convex fuzzy-number valued mappings is introduced in this study; and weighted Hermite-Hadamard type fuzzy inclusions are demonstrated for these functions. The product of two up and down $\lambda$-convex fuzzy-number valued mappings also has Hermite-Hadamard type fuzzy inclusions, which is another development. Additionally, by imposing some mild restrictions on up and down $\lambda$-convex ( $\lambda$-concave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down $\lambda$-convexity ( $\lambda$-concavity), referred to as lower up and down $\lambda$-convex (lower up and down $\lambda$-concave) and upper up and down $\lambda$-convex ( $\lambda$-concave) fuzzy number valued mappings. Using these definitions, we have amassed many classical and novel exceptional cases that implement the key findings. Our proven results expand and generalize several previous findings in the literature body. Additionally, we offer appropriate examples to corroborate our theoretical findings.