Coefficient and Fekete-Szegö problem estimates for certain subclass of analytic and bi-univalent functions

Abstract

In this paper, we obtain the Fekete-Szegö problem for the $k$-th $(k\ge 1)$ root transform of the analytic and normalized functions $f$ satisfying the condition \begin{equation*} 1+\frac{α-π}{2 \sin α}< {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\} < 1+\fracα{2\sin α} \quad (|z|<1), \end{equation*} where $\pi /2\le \alpha <\pi$. 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-Szegö problem for functions belonging to this analytic and bi-univalent function class.

9 pages