I was trying to solve the following problem:
Let $f:\mathbb D\rightarrow \mathbb D$ be holomorphic with $f(0)=0$ and $f(1/2)=0,$ where $\mathbb D=\{z:|z|<1\}.$ Then which of the following statements are correct?
(a)$|f'(1/2)|\leq 4/3,$
(b)$|f'(0)|\leq 1,$
(c)$|f'(1/2)|\leq 4/3$ and $|f'(0)|\leq 1,$
(d)$f(z)=z$ for $z\in \mathbb D.$
My Attempts: Clearly, here we can apply Schwarz Lemma.By this lemma,we can say that $|f(z)|\leq |z|$ for all $z \in\mathbb D$ and $|f'(0)|\leq 1$ and so i can say option $(b)$ is definitely correct.But i can not say anything about option$(a)$.Please help.Thanks in advance for your time.