On norm estimation for certain subclasses of analytic functions in geometric functions theory

We investigate on some subclasses of analytic fuctions defined by subordination. Also, we give estimates of $\sup_{|z| 1}\big(1-|z|^{2}\big)\big|\dfrac{f^{’’}(z)}{f^{’}(z)}\big|$, for functions belonging to extended class of starlike functions. For a locally univalent analytic function $f$ defined on $\Delta =\{z\in \mathbb{C}: |Z| 1\}$, we consider the pre-Schwarzian norm by $\Vert T\Vert=\sup _{|z| 1}\big(1-|z|^{2}\big)\big|\dfrac{f^{’’}(z)}{f^{’}(z)}\big|$. In this work, we find the sharp norm estimate for the functions $f$ in the extended classes of starlike functions.