Let $U= \{ z \in \mathbb{C} : Im z>0\}$. Let
\begin{align} u(z) = \frac{az+ b}{cz+d} \end{align} with $a,b,c,d \in \mathbb{C}$ and $u: U \rightarrow U$ one to one and onto, What i want to prove is that $a,b,c,d,$ are real.
TA says, $u(0)$ is real thus $\frac{b}{d}$ is real. But i do not understand why $u(0)$ is real.
I think $z=0=0+0i$ thus $z=0$ is not in $U$, but how can be defined? and be real?
Any other idea or method for proving $a,b,c,d,$ are real will be helpful.