What is the maximum value of f''(0) for functions in set F?

[niklas]

Let $$D\subset\mathbb{C}$$ be the unitdisc and $$F=\{f:D\rightarrow D\,|\,\forall z\in D\partial_{\bar{z}}f=0\}$$, calculate $$L=\sup_{f\in F}|f''(0)|$$. Show that there is an $$g\in F$$ with $$g''(0)=L$$.
I am a bit stuck. But I think that it might be an idea to start with Cauchy estimate. Any other ideas?

 

[niklas]

$$|a_n|\leq\frac{1}{2\pi}\frac{M}{r^3}l=\frac{M}{r^2}\quad M=\max_{|z|<r<1}|f(z)|=\sup_{z\in\partial D_r}|f(z)|$$
?

 

[niklas]

=1??