Possible values taken by an analytic function from the unit disc to the complex plane.

Question

This is a question from a competitive exam which asks for the correct option for the quetion given below.

If $f: \{z| |z|<1\} \to \mathbb{C}$ is a non constant analytic function then which of the following conditions can be possibly satisfied by f

1. $f(1/n)= f(-1/n)=1/n^2$.
2. $f(1/n) = f(-1/n^2)=1/(2n+1)$.
3. $|f(1/n)| < 2^{-n}, \hspace{.5cm} \forall n \in \mathbb{N}$.
4. $1/\sqrt{n} < |f(1/n)|< 2/\sqrt{n}, \hspace{.5cm} \forall n \in \mathbb{N}$.

So, I can see that $f(z)=z^2$ is the function which satisfies the option (1). For (2) I can use the Identity principle to show that no such function exists.

My doubt/problem is how can I show that no such function exists with the property given in (3) and (4)? The only information I can obtain is that in (3) and (4) $f(0)=0$, but I am unsure how to use this.

Any help is highly appreciated.