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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?

  1. $|f(e^{iθ})| =1$ for $0<θ<2\pi$ .
  2. $f$ maps $\{z\in\mathbb{C}: |z|<1\}$ onto itself.
  3. $f$ maps $\{z\in \mathbb{C}: |z|\le 1\}$ into itself.
  4. $f$ is one-one.

I am stuck on this problem . Can anyone help me please.................

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    i don't know where to begin..........2012-12-18

1 Answers 1