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 $delta ={zin mathbb{c}: |z|<1}$, we consider the pre-schwarzian norm by $vert tvert=sup _{|z|<1}big(1-|z|^{2}big)big|dfrac{f^{’’}(z)}{f^{’}(z)}big|$. in this work, we find the sharp norm estimate for the functions $f$ in the extended classes of starlike functions.