  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.
