  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.
