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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!!

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    $A$nd by the way solving the first inequality the second becomes pretty obvious to me...the problem is the first one!2012-06-27

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Hint: let $g = f \circ \varphi^{-1}$ where $\varphi(z) = \dfrac{z - z_0}{z - \overline{z_0}}$, and note that $g(w)/w^2$ has a removable singularity at $0$.