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  1. There is a bijective analytic function from the complex plane to the upper half-plane.
  2. There is a non-constant bounded analytic function on $\mathbb{C}\setminus\{0\}$.

Please help anyone to solve these problems.

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What are your thoughts on them? – Antonio Vargas Sep 20 '12 at 17:04

Hints :

For $1$, suppose that $f: \mathbb{C} \rightarrow \mathbb{H}$ is holomorphic, where $\mathbb{H}$ is the upper half-plane. Thus $\operatorname{Im}f(z) > 0$ for all $z \in \mathbb{C}$. Now consider $g(z):=e^{if(z)}$. Then $g$ is entire, and $|g(z)|<1$ for all $z \in \mathbb{C}$ (why?). Now what can you conclude about $g$, and then about $f$?

For $2$, Suppose that $f$ is analytic and bounded in $\mathbb{C} \setminus \{0\}$. Is it possible to extend $f$ to bounded analytic function in $\mathbb{C}$? (Hint : Riemann removable singularity theorem). Now what can you say about $f$?

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