Communications of the
Korean Mathematical Society CKMS

In our study, we present the well known family ${\mathcal N}_\Sigma^\mu(\lambda)$ of $\Sigma$, which is the family of bi-univalent functions in the open unit disk ${\mathbb U}$. We obtain sharp estimates for the first three Taylor-Maclaurin coefficients $|a_2|$, $|a_3|$, $|a_4|$. Fekete-Szeg\"{o} functional problem is also established. Furthermore, we consider the family ${\mathcal B}_{1,\Sigma}(\mu,\lambda,\gamma)$ of bi-univalent functions in the open unit disk ${\mathbb U}$. The results presented in this paper would generalize and improve some recent works of several earlier authors.

Keywords: Bi-univalent function, coefficient estimates, Fekete-Szeg"{o} problem

MSC numbers: Primary 30C45, 30C50