Trying to prove $ |f(0) | \leq a$ when $f(a)=0$ for a bounded holomorphic function $f$

Question

Suppose we have an holomorphic function $f$ such that $ |f(z) | \leq 1$ for all $z$ and suppose $f(a)=0$ ( for example $a=1/2$). I am trying to prove that $ |f(0) | \leq a$. I have tried taking the Taylor series for which the fisrt coefficient is $f(0)$ but I got nowhere with that.

Answer 1

Assuming the question is about functions on the unit disk $D=\{z:|z|\le 1\}$:

Use the Möbius transformation $h:D\to D$, $h(z)=\frac{z-a}{\bar a z-1}$ to map $h(a)=0$, $h(0)=a$. Set $g(z)=f(h(z))$ so that $g(0)=f(a)=0$. Then $\frac{g(z)}{z}$ can be holomorphically extended to $z=0$.

Now observe that $|\frac{g(z)}{z}|<1$ on $S^1=\partial D$ and use the maximum principle.