### Some General Coefficient Estimates for a New Class of Analytic and Bi-Univalent Functions Defined by a Linear Combination

#### Abstract

In the present paper, we introduce and investigate a new

class of analytic and bi-univalent functions $f(z)$

in the open unit disk $\mathbb{U}$. For this purpose,

we make use of a linear combination of the following three functions:

$$\frac{f(z)}{z}, \quad f'(z) \qquad \text{and} \qquad zf''(z)$$

for a function belonging to the normalized univalent function

class $\mathcal{S}$.

By applying the technique involving the Faber polynomials,

we determine estimates for the general Taylor-Maclaurin coefficient

of functions belonging to the analytic and bi-univalent

function class which we have introduced here. We also

demonstrate the not-too-obvious behaviour of the

first two Taylor-Maclaurin coefficients of such functions.

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