Is there exists a holomorphic function $f:D\to \bar{D}$ with $f(0)=0$ with the property $f(1/3)=1/2$???

Question

I got the question From here Existence of a holomorphic function in the open disc $D=\{z\in \mathbb{C}: |z|<1\}$...

Let $D=\{z\in \mathbb{C}: |z|<1\}$. Then there exists a holomorphic function $$f:D\to \bar{D}$$ with $f(0)=0$ with the property

1. $f'(0)=1/2$
2. $|f(1/3)|=1/4$
3. $f(1/3)=1/2$
4. $|f'(0)|=\sec{\pi/6}$

My attempts : i got option 1) and 2) correct and option 3 and option 4 is false By schawrz lemma ..but here Existence of a holomorphic function in the open disc $D=\{z\in \mathbb{C}: |z|<1\}$...... it written that option 3 is also true ,,,,im getting confused Is there exists a holomorphic function $f:D\to \bar{D}$ with $f(0)=0$ with the property $f(1/3)=1/2$???

Answer 1

A holomorphic function $f : D \to \overline{D}$ such that $f(0) = 0$ and $f \left( \frac{1}{3} \right) = \frac{1}{2}$ cannot exist from the Schwarz lemma, since it must satisfy

$$\left| f \left( \frac{1}{3} \right) \right| \leqslant \frac{1}{3}.$$