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.