I know that the bilinear map $f(z)=\frac{az+b}{cz+d},a,b,c,d\in\mathbb{R},ad−bc>0$, maps the upper half plane onto itself. But I wonder what is the map which maps lower half plane onto itself such that it is analytic and injective.Is it same as above bilnear form with different conditions on $a,b,c$ and $d?$
Analytic maps from lower half plane onto itself.
complex-analysisconformal-geometrymobius-transformation
Related Question
- [Math] Conformal maps from the unit disc onto itself, given by two sets of three points on the boundary
- [Math] Find an analytic function that maps the plane with the slit $[-1,1]$ onto the upper half plane
- [Math] Conformal maps onto open right half plane
- [Math] Conformal map onto upper-half plane.
- Analytic maps from upper half plane to itself
- Analytic function that maps upper half plane to upper half plane given two distinct values
Best Answer
A Möbius Transformation is a continuous automorphism of the extended complex plane $\Bbb C \cup \{ \infty \}$. Therefore, if $f$ maps the upper half-plane onto itself then $f$ maps also the extended real axis onto itself and the lower half-plane onto itself.
Therefore a bilinear map $f(z)=\frac{az+b}{cz+d}$ maps the upper half-plane onto itself if and only if it maps the lower half-plane onto itself.
So what you are looking for are exactly the same bilinear transformations, with the same condition $a,b,c,d\in\mathbb{R},ad−bc>0$.
(One can also argue with the Schwarz reflection principle: If $f$ maps the extended real axis onto itself then $f(\bar z) = \overline {f(z)}$ for all $z \in \Bbb C$.)