Complex analysis - Cauchy estimates

[niklas]

Homework Statement

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?

Homework Equations

The Attempt at a Solution

$$|a_2|\leq\frac{1}{2\pi}\frac{M}{r^3}l=\frac{M}{r^2}\quad M=\max_{|z|<r<1}|f(z)|=\sup_{z\in\partial D_r}|f(z)|=1?$$

 

[lippyka]

By Cauchy estimates we get|f''(0)|=|a_2|\leq\frac{1}{2\pi}\frac{M}{r^3}l=\frac{M}{r^2}=LCan I choose $g$ to be the function from Cauchy estimates?