In this paper, we obtain the Fekete-Szegö inequality for the generalized bi-subordinate functions of complex order. The various results, which are presented in this paper, would generalize those in related works of several earlier authors.******************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************

In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions using the q-derivative operator $D_q\left(0<q<1\right)$ and the Gegenbauer polynomials in a symmetric domain, which is the open unit disc $\mathsf{\Lambda }=\left\{\wp :\wp \in \mathbb{C}\mathrm{and}\left|\wp \right|<1\right\}.$ For these subclasses of analytic and bi-univalent functions, the coefficient estimates and Fekete–Szegö inequalities are solved. Some special cases of the main results are also linked to those in several previous studies. The symmetric nature of quantum calculus itself motivates our investigation of the applications of such quantum (or q-) extensions in this paper.