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.

This study established upper bounds for the second and third coefficients of analytical and bi-univalent functions belonging to a family of particular classes of analytic functions utilizing $q-$Ultraspherical polynomials under $q-$Saigo's fractional integral operator. We also discussed the Fekete-Szegö family function problem. As a result of the specialization of the parameters used in our main results, numerous novel outcomes were demonstrated.

In this paper, we consider a subclass of bi-univalent functions, denoted by ${\mathbb{C}\mathbb{S}}_{\beth }^{\ast }(\gamma )$, which is defined via a balloon-shaped domain associated with the function $\frac{2\sqrt{1+\varsigma }}{1+e^{-\varsigma }}$. We prove that this class is non-empty using illustrative mappings. We investigate upper bounds for the second- and third-order Hankel determinants, focusing on the functional $H_3(1)$. In addition, we estimate the Taylor coefficients up to order $l=5$, which are essential in obtaining bounds for the determinants. The balloon shape introduces new analytic features that enrich the behavior of the functions in this class. Several examples and special cases are offered to illustrate the flexibility of the outcomes.