Improved Version of Hermite-Hadamard Inequalities by Generalized Fuzzy Fractional Operators in Aspects of Convexity

Abstract

Special functions play a crucial role in the development of fractional calculus and have extensive applications in solving problems involving differential equations, integrals, and series solutions. In this thesis, we introduce two-dimensional Riemann-Liouville fractional operators, Riemann-Liouville fuzzy fractional operators (FFOs), and gener- alized fuzzy fractional operators incorporating the extended Bessel-Maitland function (E¹vBMF) as a kernel. We also construct two-dimensional h-Godunova-Levin convex functions, coordinated pre- invex functions, and various types of fuzzy convex and non- convex functions. Furthermore, we derive new forms of Hermite-Hadamard (H-H) and trapezoid-type fractional integral inequalities for these newly defined two-dimensional convex and pre-invex functions. We also examine the existence and generalization of H-H and trapezoid-type inequalities using the framework of fuzzy interval-valued functions (FIVFs), which extend classical interval- and real-valued functions. By effectively applying generalized fuzzy fractional operators with (E¹vBMF) kernels, we establish new  inequalities and discuss additional applications arising from our main results, presented in the form of corollaries.

Keywords: Riemann–Liouville fractional operators; Fuzzy fractional operators (FFOs); Extended Bessel–Maitland function (E1vBMF); h-Godunova–Levin convex functions; Pre-invex functions; Fuzzy interval-valued functions (FIVFs).