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 $\pi/2\leq \alpha<\pi$. Afterwards, by the above two-sided inequality we introduce and investigate 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 belonging to this analytic and bi-univalent function class.
Categories: math.CV