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I recently came across the following problem

Pick out the true statements:

  • There is a bijective analytic function from $\Bbb C$ to the upper half plane $\Bbb H$

  • There is a non-constant bounded analytic function on $\Bbb C \setminus \{0\}$

  • If $\{a_n\}$ and $\{b_n\}$ are two sequences of positive real numbers with $a_n \longrightarrow 0$ and $b_n$ diverging to $\infty$, then the sequence $c_n=a_ne^{ib_n} \longrightarrow 0$


My try

For the first bullet, the statement is false, since if it is true, then the range of such a function is dense in $\Bbb C$ but $\Bbb H$ is not dense in $\Bbb C$.

For the third one, the statement is true. Since $$c_n \longrightarrow 0 \iff (\Re( c_n) \longrightarrow 0) \wedge (\Im(c_n) \longrightarrow 0)$$ The above is true, since $$\vert \Re (c_n) \vert=\vert a_n \cos b_n\vert \leq \vert a_n \vert \to 0$$ and $$\vert \Im (c_n) \vert=\vert a_n \sin b_n\vert \leq \vert a_n \vert \to 0$$

Is this correct? May I have a hint for the second one?

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The third one is simpler than that. Just use the fact that $\lvert a_ne^{ib_n}\rvert=a_n$.

For the second one, think about the type of singularity that such a function would have at $0$.

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  • $\begingroup$ Thanks! How about the first one? Is my reasoning correct? $\endgroup$ – Chinnapparaj R Feb 10 at 16:24
  • $\begingroup$ Yes, it is correct. $\endgroup$ – José Carlos Santos Feb 10 at 16:25
  • $\begingroup$ In the third one, we cannot use the divergence of $b_n$ . Is this divergence assumption, not needed here ? $\endgroup$ – Chinnapparaj R Feb 10 at 16:34
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    $\begingroup$ I had no use for it. $\endgroup$ – José Carlos Santos Feb 10 at 16:35

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