$F=\{ f: \mathbb{C} \to \mathbb{C}| f$ is an entire function,$ |f'(z)| \leq |f(z)|$ for all $z \in \mathbb{C}\}$

Question

Consider $F=\{ f: \mathbb{C} \to \mathbb{C}| f$ is an entire function,$ |f'(z)| \leq |f(z)|$ for all $z \in \mathbb{C}\}.$

Then which of the following is true?

1) $F$ is a finite set.

2) $F$ is an infinite set.

3) $F=\{ \beta e^{\alpha z}: \beta \in \mathbb{C}, \alpha \in \mathbb{C}\}$

4) $F=\{ \beta e^{\alpha z}: \beta \in \mathbb{C}, |\alpha| \leq 1\}$

1) is false and 2) is true as functions in option 4) satisfy above propetry.

Also 3) is false as for $\alpha=2 $ and $\beta =1$ the property does not hold true.

Is 4) true?