Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
2026-03-27 16:37:48.1774629468
holomorphic functions with nonvanishing derivative on unit disk $D$
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No, take something like $f(z) = e^{10z}$. Then $f(0)=f(2\pi i/10)$, but $f'(z) \neq 0$ everywhere. (Assuming that you are asking whether $f$ has to be injective.)