An Application of Liouville's theorem for entire function

Question

Let $f:\mathbb{C} \to \mathbb{C}$ be entire function. Suppose that f=u+iv where u and v are real and imaginary parts of f respectively. f is constant if

1. $\{u(x,y)|z \in \mathbb{C}\}$ is bounded.
2. $\{v(x,y)|z \in \mathbb{C}\}$ is bounded.
3. $\{u(x,y)+v(x,y)|z \in \mathbb{C}\}$ is bounded.
4. $\{u^2(x,y)+v^2(x,y)|z \in \mathbb{C}\}$ is bounded.

For (1) and (2) if we take $e^{f(z)}$ and $e^{-if(z)}$ then they are bounded and entire implies that f(z) is constant. Please give me some idea about (3) and (4). Thanks in advance.