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


1 Answer

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  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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