existence of holomorphic functions Let $D$= { $z \epsilon \Bbb C :|z|<1$ } .   Then there exists a holomorphic function $f:D\to \overline D$ with $f(0)=0$ with the property
a) $f'(0)={1\over 2}$
b) $|f(1/3)|={1\over 4}$
c) $f(1/3)={1\over 2}$
d) $|f'(0)|=\sec(\pi/6)$

when $f(z)={1\over 2}z$ then $a)$ is true
when $f(z)={1\over 4}$ then $b)$ is true
To solve the problem do we've to take such examples?Or is there any other way?


Hint:
         If $f(z)$ is holomorphic function from $D$ onto itself, where $D=\{z\in\mathbb{C}:|z|<1\}$ and $f(0)=0$, then $|f(z)|\le|z|$ and $|f'(z)|\le1$ for all $z\in D$, by Schwartz lemma
