What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$?

Question

let $f:D=\{z\in \mathbb C:|z|<1\} \to \overline D$ with $f(0)=0$ be a holomorphic function.

What will be $f^{'}(0)$ and $f(\dfrac{1}{3})$?

My try:By cauchy integral formula : $f^{'}(0)=\dfrac{1}{2\pi i}\int_\gamma \dfrac{f(z)}{z^2}dz$ where $\gamma$ is a simple closed positively oriented contour and $f$ is analytic in $\gamma$.

Since $f(0)=0$ so $f(0)=\ \dfrac{1}{2\pi i}\int_\gamma \dfrac{f(z)}{z}dz=0 \implies \int_\gamma \dfrac{f(z)}{z^2}dz$

But how should I use this calculate the above.Any help

Answer 1

There is no unique answer. You do however, have the Schwarz Lemma to make the estimates $|f'(0)| \le 1$ and $|f(1/3)| \le 1/3$.

Answer 2

Observe that this conditions are not enough to find these values, because we can take $$ f(z)=\omega z $$ where $\omega$ is an arbitrary complex constant with $|\omega|\le1$.

In questions like this you should try to use Schwartz lemma to apply.