Communications of the
Korean Mathematical Society CKMS

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 $E=\left\{ z:\left\vert z\right\vert <1\right\} $ and on $\partial E=\left\{ z:\left\vert z\right\vert =1\right\} $. For the function $ f(z)=z^{2}+c_{3}z^{3}+c_{4}z^{4}+\cdots$ defined in the unit disc $E$ such that $f(z)\in \mathcal{A}_{\alpha }$, we estimate a modulus of the angular derivative of $\frac{zf^{\prime }(z)}{f(z)}$ function at the boundary point $b$ with $\frac{bf^{\prime }(b)}{f(b)}=1+\alpha $. Moreover, Schwarz lemma for class $\mathcal{A}_{\alpha }$ is given. The sharpness of these inequalities is also proved.

Keywords: Schwarz lemma, Jack's lemma, angular limit

MSC numbers: Primary 30C80, 32A10