Property of holomorphic function from open unit disc to open unit disc

Question

This question was asked in masters entrance exam for which I am preparing .

Question : Let $f\colon\mathbb{D} \to \mathbb{D} $ be an holomorphic function with $f(0)=0$, where $\mathbb{D} $ is open unit disc $\{z \in\mathbb{C} :|z| < 1 \}$ . Then which one of the following is true.

1. $|f'(0)|=1$.
2. $|f(1/2)|\leq1/2$.
3. $|f(1/2)|\leq1/4$.
4. $|f'(0)|\leq1/2$.

I am really confused and I have no idea on what should be done so I am unable to show an attempt.

Answer 1

This is just an application of the Schwarz lemma. No. 2 is true, as directly stated in the lemma. For the others, $f(z)=0$ or $f(z)=z$ are counterexamples.