On the symmetric $q$-analog on the bi-univalent functions with respect to symmetric points

Our objective is to usher and investigate the subclass$\widetilde{\mathcal{S^{*}_{\sum}}}^{\eta}_{q}(\mu,\lambda;\phi)$ of the function class $\sum$ of analytic and bi-univalent functions related with the symmetric $q$-derivative operator and the generalized Bernardi integral operator. On the one hand, without the generalized Bernardi integral operator we estimate the second Hankel determinants for the reduced subclasses $\widetilde{\mathcal{S^{*}_{\sum}}}_{q}(\lambda;\phi)$ with respect to symmetric points. On the other hand, we also give the corresponding results of Fekete-Szeg\"{o} functional inequalities and the upper bounds of the coefficients $a_2$ and $a_3$ for these subclasses.