Prove that every holomorphic function on the closed disk $\overline{\Delta}(0,1)$ with $|f(z)|<1$ when $z\in \overline{\Delta}(0,1)$ has at least one fixed point in $\Delta (0,1)$.
I was thinking that I could use Maximum Modulus Principle for function $g = \dfrac{1}{f(z)-z}$, but I got stuck at here (because I can't have any argument from here). Can anybody help me?
P/s: I've read two solutions in here but I can't use Rouche theorem (we can only use Maximum Modulus Principle), and the next solution I think that was wrong (because if he applies the Cauchy's estimate, he only has $|f'(z)| \le \dfrac{r}{m}$, with $m \rightarrow 0$ since $|z| \rightarrow 1$.) Can anybody help me to complete that solutions? Thanks.
(Sorry but I can't ask in the above post according to mathstackexchange's rule.)