Let $U$ be an open set of $\mathbb{C}$ containing $D=\{z\in \mathbb{C}: |z|<1\}$ and let $f:U\to \mathbb{C}$ be map defined by $f(z)= e^{iψ} \frac{z-a}{1-\overline{a}z}$ for $a\in D$ and $ψ\in[0,2\pi]$. Which of the following statements are true?
- $|f(e^{iθ})| =1$ for $0<θ<2\pi$ .
- $f$ maps $\{z\in\mathbb{C}: |z|<1\}$ onto itself.
- $f$ maps $\{z\in \mathbb{C}: |z|\le 1\}$ into itself.
- $f$ is one-one.
I am stuck on this problem . Can anyone help me please.................