Unified sufficient conditions for generalized classes of analytic functions associated with the Rabotnov fractional operator

In this paper, we investigate the analytical landscape of the normalized Rabotnov function together with certain linear and integral operators acting on subclasses of analytic and univalent functions with negative coefficients. By leveraging convolution techniques and coefficient-based criteria, we derive innovative sufficient requirements for the function to be categorized under the subclasses $P_{{\alpha }_3}^{\ast }({\alpha }_1,{\alpha }_2)$ and ${\mathrm{\wp }}_{{\alpha }_3}^{\ast }({\alpha }_1,{\alpha }_2)$. Particular attention is given to the role of symmetry in the associated operators, which provides additional structural insight into the behavior of these analytic functions. The study specifically explores the interaction between Rabotnov-derived operators and analytic functions characterized by real-part inequalities, thus providing a comprehensive analysis of their inclusion properties. We present a series of corollaries that extend these properties to classical starlike and convex functions of order ${\alpha }_1$, thus broadening the scope of previous investigations. Furthermore, we provide analogous geometric results for a specialized integral operator, thus reinforcing the potential for these techniques to be applied to broader families of special functions in mathematical physics.