# Questions tagged [mobius-transformation]

For questions about the geometry, complex-analytic, and group-theoretic properties of the Möbius transformations (linear fractional transformations) $z \mapsto \frac{az + b}{cz + d}$ of the complex plane, which can be identified with the group $PGL(2, \mathbb{C})$, or certain subgroups thereof.

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### Is a rational function which maps all circles/lines to circles/lines a Möbius transformation?

It is well-known that Möbius transformations map circles and lines to circles and lines. (Here and in the following, “line” means a line in the extended complex plane $\hat{\Bbb C}$, including the ...
### If $f$ is holomorphic in $D = \Bbb D\cap\{\Re(z)>0\}$, $f(a)=a$ for $a\in D$ and $f(D)\subset D$, how to show that $|f'(a)|\le 1$?
Consider $D = \{z\in\Bbb C : |z|<1,\;\Re(z)>0\}$. Take $a\in D$ and consider $f$ a holomorphic function in $D$ such that $f(a)=a$ and $f(D)\subset D$. How can we prove that $|f'(a)|\le 1$? I've ...