### 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.

#### Keywords

Analytic functions; Univalent functions; Taylor-Maclaurin series representation; Faber polynomials; Bi-Univalent functions; $q$-Derivative operator; $q$-hypergeometric functions; $q$-Integral operators.

#### Full Text:

PDFDOI: http://dx.doi.org/10.24193/subbmath.2018.4.01

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