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(a) Let $H = \{z ∈ \Bbb C : \text{Im}(z) > 0\}$ be the (open) upper half-plane. Define a map $f : H → \Bbb C$ by: $$f(z) =\frac{z-i}{z+i}$$

Show that $f$ is analytic, $|f(z)| < 1$ for all $z$ and, finally, if $|w| < 1$, then $f(z) = w$ for some $z ∈ H$.

(b) In contrast, show that there is no analytic map $g : \Bbb C → D(0, 1)$, such that whenever $|w| < 1$, there exists $z ∈ \Bbb C$ satisfying $g(z) = w$.

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closed as not constructive by Did, Andrés Caicedo, tomasz, TMM, J. M. Nov 14 '12 at 8:35

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

Nine questions, multiple proddings on several of them to add ANYTHING AT ALL personal to these, and yet... – Did Nov 13 '12 at 22:55
up vote 1 down vote accepted
  1. Using the fact that the real part of $z$ is positive, calculate the real and imaginary parts of $f(z)$; that is, find $a,b \in \mathbb{R}$ (in terms of the real and imaginary parts of $z$) such that $f(z) = a + ib$. Then you merely need to show $a^2 + b^2 < 1$. From here it should be possible to argue that $a,b$ can achieve any values on the open unit disc.

  2. If $g: \mathbb{C} \rightarrow D(0,1)$ is analytic, then $g$ is bounded and entire. What do you know about bounded and entire analytic functions?

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