Applications of the Jack's lemma for the holomorphic functions

Selin Aydınoğlu, Bülent Nafi Örnek

In this paper, we give some results on $\frac{zf^{\prime }(z)}{f(z)}$ for the certain classes of holomorphic functions in the unit disc $U=\left\{ z:\left\vert z\right\vert <1\right\} $ and on $\partial U=\left\{ z:\left\vert z\right\vert =1\right\} $. For the function $% f(z)=z+c_{2}z^{2}+c_{3}z^{3}+...$ defined in the unit disc $U$ such that $f\in\mathcal{M}$, we estimate a modulus of the angular derivative $\frac{% zf^{\prime }(z)}{f(z)}$ function at the boundary point $b$ with $f^{\prime }(b)=0$. Moreover, Schwarz lemma for class $\mathcal{M}$ is given. The sharpness of these inequalities is also proved.