Fekete-Szegö inequality for a subclass of analytic functions associated with Gegenbauer polynomials

Abstract

In this paper, we define a subclass of analytic functions by denote $T_{\beta}H\left( z,C_{n}^{\left( \lambda \right) }\left( t\right) \right)$ satisfying the following subordinate condition \begin{equation*} \left( 1-\beta \right) \left( \frac{zf'\left( z\right) }{f\left( z\right) }\right) +\beta \left( 1+\frac{zf^{\prime \prime}\left( z\right) }{f'\left( z\right) }\right) \prec \frac{1}{\left( 1-2tz+z^{2}\right) ^{\lambda }}, \end{equation*} where $\beta \geq 0$, $\lambda \geq 0$ and $t\in \left( \frac{1}{2},1\right] $. We give coefficient estimates and Fekete-Szegö inequality for functions belonging to this subclass.

Keywords

• analytic and univalent function
• typically real function
• subordination
• gegenbauer polynomial
• coefficient estimate
• fekete-szegö inequality