Existence of a holomorphic function in the open disc $D=\{z\in \mathbb{C}: |z|<1\}$. Let $D=\{z\in \mathbb{C}: |z|&lt;1\}$. Then there exists a holomorphic function $$f:D\to \bar{D}$$ with $f(0)=0$ with the property 


$f'(0)=1/2$
$|f(1/3)|=1/4$
$f(1/3)=1/2$
$|f'(0)|=\sec{\pi/6}$


I have no idea how to come up with such a fuction, which satisfies atmost two of the above conditions as there are contradictory statements.  I tried to guess some but it does not work. Is there any general rule to solve this kind of problems? Which options are true? Please help me out. Any hint is sufficient.

edit - Answer says $1$ and $2$ are correct.
By Schwarz lemma, $1$, $2$ and $3$ are true trivially.
I think 3rd is not true. As by Schwarz lemma $|f(z)|\leq|z|$ which is not true in 3rd option