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?