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.