prove or give counter example, for every holomorphic function on the unit disc there is $f(z)=z$

Question

let f be a holomorphic function on $D={zin mathbb C:|z|<1}$. and let $f$ be continuous on $cl(D)$ and $f[D]subseteq D$.
Prove or give counter example,
$exists zin Dmathrm{.f(z)=z}$

Pythagoras · Accepted Answer

Counter example: Let $a$ with $0<|a|<1$ and $$f(z)=frac{z-a}{1-bar a z}.$$

prove or give counter example, for every holomorphic function on the unit disc there is $f(z)=z$

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