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I am going through my weekly assignment of complex analysis but I am stuck at two points, since I don't understand a condition in the first case and what I'm being asked in the second one.

In the first part of the assignment I am given an entire function $f : \mathbb{C} \to \mathbb{C}$ such that

$$|f(z)| \leq \frac{1}{|\mathfrak{Im}(z)|}, \qquad \forall z \in \mathbb{C}$$

I have to prove that $f(z) = 0, \ \forall z \in \mathbb{C}$.

What is supposed to happen when $\mathfrak{Im}(z) = 0$? We are not working in $\overline{\mathbb{C}}$, so we don't assume that $f(z) = \infty$ for some $z \in \mathbb{C}$. How should I interpret this condition?

In another part of the assignment I am told to classify entire bi-periodic functions. I know what is meant by classification when one is talking about e.g. manifolds or surfaces, but I am finding it hard to interpret in this case. At least this follows just from the definition:

Let $f$ be an entire bi-periodic function with periods $p_1, p_2 \in \mathbb{C}$, such that $p_1, p_2$ are $\mathbb{R}$-linear independent. Then we have

$$f(z + jp_1 + kp_2) = f(z) \qquad \forall z \in \mathbb{C}, \ \forall j, k \in \mathbb{Z_{\geq0}}$$

How should I categorize these functions?

eslukas
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  • It might be important which theorem your first question relates to. For instance, perhaps $f$ is only defined on the upper half plane. In any case, adding the context will help people answer your question. – carmichael561 May 18 '17 at 19:11
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    The way I would read the inequality $\lvert f(z)\rvert\le\dfrac{1}{\lvert \Im z\rvert}$ for $z\in\Bbb R$ would be "no constraint". That being said, I might be wrong. –  May 18 '17 at 19:17
  • I have added the full assignment description, thanks for telling me! – eslukas May 18 '17 at 19:17
  • What @G.Sassatelli says. It's no constraint on $\mathbb{R}$, only on $\mathbb{C}\setminus \mathbb{R}$. – Daniel Fischer May 18 '17 at 19:53
  • If $g$ is continuous and doubly-periodic then it is bounded. If it is entire then it is constant. – reuns May 18 '17 at 20:08
  • These are two unrelated questions. Please post such separately. Also, both are (frequent) duplicates. See for example: https://math.stackexchange.com/questions/1350856 or https://math.stackexchange.com/questions/377782 for the first, and https://math.stackexchange.com/questions/1244416 just to mention one. – mrf May 18 '17 at 20:49
  • @mrf I don't think these are duplicates, because OP doesn't ask for a solution, but rather an explanation of what's required. Pointing at an answer doesn't help with that (in fact, it defeats the purpose). – Jonathan Y. May 18 '17 at 21:33

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