Norm estimates of the pre--schwarzian derivatives for two certain subclasses of starlike functions

Abstract

Let $\alpha\in[\pi/2,\pi)$ and $\gamma_1,\gamma_2\in(0,1]$. For a normalized analytic functions $f$ in the open unit disc $\Delta$ we consider\begin{equation*}%\label{1definition}  \mathcal{M}(\alpha):=\left\{f\in\mathcal{A}:  1+\frac{\alpha-\pi}{2 \sin \alpha}<    {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\} <    1+\frac{\alpha}{2\sin \alpha}, \quad z\in\Delta\right\},\end{equation*}and\begin{equation*}%\label{1definition}  \mathcal{S}^*_t(\gamma_1,\gamma_2):=\left\{f\in\mathcal{A}: -\frac{\pi\gamma_1}{2}< \arg\left\{\frac{zf'(z)}{f(z)}\right\}    <\frac{\pi\gamma_2}{2}, \quad z\in\Delta\right\}.\end{equation*}In the present paper, we establish a sharp norm estimate of the pre--Schwarzian derivative for functions $f$ belonging two these subclasses of analytic and normalized functions.