In this work, we address and explore the concept of generalized \(m\)-preinvex functions on fractal sets along with linked local fractional integral inequalities. Additionally, some engrossing algebraic properties are presented to facilitate the current initiated idea. Furthermore, we prove the latest variant of Hermite-Hadamard type inequality employing the proposed definition of preinvexity. We also derive several novel versions of inequalities of the Hermite-Hadamard type and Fejér-Hermite-Hadamard type for the first-order local differentiable generalized \(m\)-preinvex functions. Finally, some new inequalities for the generalized means and generalized random variables are established as applications.

This article introduces certain algebraic properties of generalized $({\tilde{h}}_1,{\tilde{h}}_2)$-pre-invex functions on ${\mathbb{R}}^{\mathsf{\aleph }}$$(0<\mathsf{\aleph }\le 1)$. A new fractal weighted integral identity is established and further employed to obtain several Ostrowski-type results in the fractal setting for functions whose first derivatives in the modulus belong to the generalized $({\tilde{h}}_1,{\tilde{h}}_2)$-pre-invex functions’s class. An illustrative example is presented to validate the theoretical findings. Moreover, applications of the main results are derived in connection with generalized random variables and various special means, highlighting the effectiveness and potential scope of the proposed approach.