For a non-constant entire function which property is possible? Let $f$ be a non-constant entire function.Which of the following properties is possible for each $z \in \mathbb{C}$
$(1) \  \ \mathrm{Re} f(z) =\mathrm{Im} f(z)$
$(2) \ \ |f(z)|<1$
$(3)\ \ \mathrm{Im} f(z)<0$
$(4)\ \  f(z) \neq 0 $
I tried for $(2)$ and $(3)$ option.For $(2) f$ is entire and bounded by Louiville's theorem it has to be constant which is contradiction to hypothesis.for $(3)$ if imaginary part or real part is bounded below or above then function has to be constant.How eliminate $(1)$ & $(4)$? I don't know what are the right option. 
Here it is possible that there are more than one answers. Please help me thanks in advance.

Hints: For $1,$ note that $f$ can't be $0$ everywhere, nor can its derivative. Hence, there is some non-empty open set that $f$ maps to an open set. (Why?) Can the line $\operatorname{Re}(w)=\operatorname{Im}(w)$ contain any non-empty open set?
For $4,$ try to think of an example of a non-constant entire function that is never $0.$ (A basic example should do.)
