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

Our objective is to usher and investigate the subclass${\widetilde{{\mathcal{S}}_{\sum }^{\ast }}}_q^{\eta }(\mu ,\lambda ;\phi )$ 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 ${\widetilde{{\mathcal{S}}_{\sum }^{\ast }}}_q(\lambda ;\phi )$ with respect to symmetric points. On the other hand, we also give the corresponding results of Fekete-Szegö functional inequalities and the upper bounds of the coefficients $a_2$ and $a_3$ for these subclasses.