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

Mathematics Asked by hash man on August 4, 2020

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,
$$exists zin Dmathrm{.f(z)=z}$$

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

Correct answer by Pythagoras on August 4, 2020

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