Let $f$ be a holomorphic function from $H=\{z \in \mathbb C : \operatorname{Im}(z)>0\}$ to $D=\{z \in \mathbb C : |z|<1\}$. Suppose that $f(z_0)=f'(z_0)=0$.
Show that: \[ |f(z)| \leq \frac{|z-z_0|^2}{|z-\overline{z}_0|^2} \quad\text{and}\quad |f''(z_0)| \leq \frac{1}{2|\operatorname{Im}(z_0)|^2} \]
Thanks everyone for your help!!