Could anyone tell me what the difference is between a map which is conformal, bi-holomorphic and an automorphism from $D\rightarrow D$ or $D$ to the upper half plane (in that case I know that is not automorphism)?
Maybe I am getting confused about terminology? Please someone explain with examples.
1 Answer
A conformal map is a holomorphic map whose derivative does not vanish. So it must be locally injective, but not necessarily surjective or injective.
A biholomorphism is a map which is bijective and holomorphic (then its inverse is also holomorphic). An automorphism is a biholomorphism $U \rightarrow U$ where $U$ is a complex domain (or a Riemann surface). In your example they are the same thing.
Example of a conformal map which is not injective : $z \mapsto e^z$. Its derivative does not vanish but it is not injective (and not surjective since its range doesn't contain $0$).
About the automorphism of the unit disk : it can be shown that they are exactly the : $B_\alpha(z)=\frac{\alpha -z}{1-\overline{\alpha}z}$, where $|\alpha|<1$.
