Coefficient and Fekete-Szeg\"{o} problem estimates for certain subclass of analytic and bi-univalent functions

Abstract

In this paper, we obtain the Fekete-Szeg\"{o} problem for the $k$-th $(k\geq1)$ root transform of the analytic and normalized functions $f$ satisfying the condition
\begin{equation*}
1+\frac{\alpha-\pi}{2 \sin \alpha}<
{\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\} <
1+\frac{\alpha}{2\sin \alpha} \quad (|z|<1),
\end{equation*}
where $\alpha\in[\pi/2,\pi)$. Afterwards, by the above two-sided inequality we introduce a certain subclass of analytic and bi-univalent functions in the disk $|z|<1$ and obtain upper bounds for the first few coefficients and Fekete-Szeg\"{o} problem for functions $f$ belonging to this class.