Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities

Abstract

The theory of convex mapping has a lot of applications in the field of applied mathematics and engineering. The fuzzy Riemann-Liouville fractional integrals are the most significant operator of fractional theory which permits to generalize the classical theory of integrals. This study considers the well-known Hermite-Hadamard type and associated inequalities. To full fill this mileage, some new versions of fuzzy Hermite-Hadamard type and Hermite-Hadamard-Fejér type inequalities for up and down convex fuzzy-number valued mappings have been obtained. Some new related fuzzy Hermite-Hadamard type inequalities are also obtained with the help of product of two up and down convex fuzzy-number valued mappings. Moreover, we have introduced some new important classes of fuzzy numbered valued convexity which are known as lower up and down convex (concave) and, upper up and down convex (concave) fuzzy numbered valued mappings by applying some mild restrictions on up and down convex (concave) fuzzy numbered valued mappings. By using these definitions, we have acquired many classical and new exceptional cases which can be viewed as applications of the main results. We also present three examples of fuzzy numbered valued convexity to demonstrate the validity of the fuzzy inclusion relations proposed in this paper.