The Hadamard and Generalized Fractional Integral Fuzzy-Number-Valued Operators for Mappings of One and Two Variables, and Their Related Fuzzy Number Inequalities

In this study, we introduce new versions of fuzzy fractional integral operators for both one- and two-variable cases. Using these operators, several Hermite–Hadamard-type (H-type) inclusions are established for fuzzy-number-valued convex functions (F·N⋅V-functions) and F·N⋅V-coordinated convex functions. These results are obtained by employing F·N⋅V-weighted functions within the framework of the newly defined Hadamard and generalized fractional integrals in one- and two-dimensional settings. The use of generalized fractional integral operators provides a unified approach that encompasses a wide class of classical and modern fractional integrals, including the fuzzy Riemann–Liouville and Hadamard types. This unified setting enables the derivation of more comprehensive and flexible inequality results in the fuzzy-number context. The inclusions obtained in this work significantly extend and generalize several known H⋅H-type inequalities previously established for real-valued and interval-valued functions (I⋅V-functions). Furthermore, the proposed results yield a variety of meaningful special cases by specifying suitable kernel functions and parameters of the generalized fractional integrals. In particular, we derive new weighted H⋅H-type inclusions involving logarithmic functions in the fuzzy-number framework. These findings underscore the effectiveness of generalized fractional integrals in capturing nonlocal behavior and uncertainty, and they provide new tools for further investigations in fuzzy analysis, fractional calculus, and generalized convexity.