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?


Another approach:
                 You can think of a linear function f(z)=az, where |a|≤1 for all z in D  and satisfying f(0)=0. Then f '(z)=a ,|a|≤1.
You can easily see that options (3) and (4) do not satisfy the condition |a|≤1.
