Let $\mathbb{D}= \{ z \in \mathbb{C} : |z| <1\}$. Which of the following are correct?
1) There exists a holomorphic function $f: \mathbb{D} \to \mathbb{D}$ s.t $f(0)=0$ and $f'(0)=2$.
2) There exists a holomorphic function $f: \mathbb{D} \to \mathbb{D}$ s.t $f(\frac{3}{4})=\frac{3}{4}$ and $f'(\frac{2}{3})=\frac{3}{4}$.
3) There exists a holomorphic function $f: \mathbb{D} \to \mathbb{D}$ s.t $f(\frac{3}{4})=\frac{-3}{4}$ and $f'(\frac{3}{4})=\frac{-3}{4}$.
4) There exists a holomorphic function $f: \mathbb{D} \to \mathbb{D}$ s.t $f(\frac{1}{2})=\frac{-1}{2}$ and $f'(\frac{1}{4})=1$.
1) is not true by Schwarz lemma. How to comment on 2),3) and 4)?
I think 3) is also true as $|f'(\frac{3}{4})| \leq \dfrac{1-|f(\frac{3}{4})|^2}{1-|\frac{3}{4}|^2}$ holds.
