This study introduces for the first time a class of generalized superquadratic functions specifically on fractal sets and explores their unique features. The research develops several generalized inequalities, including Jensen’s, converse Jensen’s, Mercer Jensen’s and Hermite-Hadamard’s inequalities based on the properties of generalized superquadratic functions. The findings are confirmed through reduced results, numerical calculations and graphical depictions, ensuring the robustness and accuracy of the proposed inequalities by taking into account several appropriate examples. A detailed comparative analysis between inequalities derived from generalized superquadratic functions and those from generalized convex functions, highlighting the greater refinement provided by the generalized superquadratic functions. The study enhances its findings with practical applications in probability expectations and special means in fractal space, demonstrating the applicability and relevance of the new results in these domains. The new results presented in this work provide significant extensions and improvements over existing literature, showcasing advancements and potential for further research in the field.

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by $\alpha$, these operators offer a unique flexibility: setting $\alpha =1$ recovers the classical inequalities for harmonically convex functions, while setting $\alpha =0$ yields inequalities for differentiable harmonically convex functions. This framework allows us to unify classical and fractional cases within a single operator. To validate the theoretical results, we provide several illustrative examples supported by graphical representations, marking the first use of such visualizations for inequalities derived via P.C.H operators. Additionally, we demonstrate practical applications of the results by deriving new fractional-order recurrence relations for the modified Bessel function of type-1, which are useful in mathematical modeling, engineering, and physics. The findings contribute to the growing body of research in fractional inequalities and harmonic convexity, paving the way for further exploration of generalized convexities and higher-order fractional operators.