Existence of a holomorphic function with the desired property Let $D=\{z\in \mathbb C:|z|<1\}$.
Then does there exist 
1.a holomorphic function $f:D\to \overline D$ with  $f(0)=0$ and $|f(\frac{1}{3})|=\frac{1}{4}$ ?
2.a holomorphic function $f:D\to \overline D$ with  $f(0)=0$ and $f(\frac{1}{3})=\frac{1}{2}$ ?
How to approach these?I am new to complex analysis.So please do provide some hints or the theorems that can help me here?

For 1,  use $\frac{3}{4}·\frac{1}{3} = \frac{1}{4}$ and $\lvert\frac{3}{4}\rvert < 1$.
For 2, use the open mapping theorem and the Schwarz lemma.
