The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain $q$-integral operator

Abstract

In our present investigation, we first introduce several new subclasses of
analytic and bi-univalent functions by using a certain $q$-integral
operator in the open unit disk
$$\mathbb{U}=\{z: z\in \mathbb{C} \quad \text{and} \quad \left
\vert z\right \vert <1\}.$$
By applying the Faber polynomial expansion
method as well as the $q$-analysis, we then
determine bounds for the $n$th coefficient in the
Taylor-Maclaurin series expansion for functions
in each of these newly-defined analytic and
bi-univalent function classes subject to a gap series condition.
We also highlight some known consequences of our main results.