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|<۱}\big(۱-|z|^{۲}\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|<۱\}, we consider the pre-Schwarzian norm by \Vert T\Vert=\sup _{|z|<۱}\big(۱-|z|^{۲}\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.

Keywords:

Analytic Functions , Starlike functions , Pre-schwarzian derivatives , Subordination