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Let $U$ be an open subset of $\mathbb{C}$ containing $\{z\in\mathbb{C}\mid |z|\leq 1\}$ and let $f:U\to\mathbb{C}$ be the map defined by $f(z)=e^{i\omega}(z-a)/(1-\overline{a}z)$ for $a\in D$ and $\omega\in [0,2\pi]$.
Which of the following are true?

(a) $|f(e^{i\theta})|=1$ for $0≤\theta≤ 2\pi $
(b) $f$ maps $\{z\in\mathbb{C}\mid|z|\leq1\}$ onto itself
(c) $f$ maps $\{z\in\mathbb{C}\mid|z|\leq 1\}$ into itself
(d) $f$ is one-one

How should I able to solve this problem. Can anyone help me please

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    What is $D$? The unit circle?2012-12-16
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    The unit circle or the unit disc?2012-12-16
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    extremely sorry.it is open unit disk.2012-12-16
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    This is almost certainly a homework and was asked recently on the site. // OP: What are your thoughts on the question, what do you know on the subject, what did you try to solve this problem?2012-12-16
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    OP: Whether you are [bdas](http://math.stackexchange.com/q/223713) or not, your definition of $f$ is plagued by the same problem than theirs: what is the image of $1/\bar a$ when $1/\bar a$ is in $U$?2012-12-16

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