To solve this problem I thought about using the function $g(z) = \frac{f(z)}{z}$ because I know it is also holomorphic, and then to apply on $g(z)$ the Maximum Modulus Principle, which I recall below:
If $f$ is holomorphic in a domain D and continuous on it, then $|f(0)| \leq \max_{z \in D} |f(z)|$
I know it can be done this way, but I don't understand how to apply this Principle since $g(0)$ is actually not defined.. any clue?
Best Answer
Define $g(z):=\begin{cases} \frac{f(z)}{z}&\text{if $z\neq 0$}\\ f'(0) & \text{if $z=0$}\end{cases}$. Now, note that since $f(0)=0$, we have \begin{align} \lim\limits_{z\to 0}g(z)=\lim_{z\to 0}\frac{f(z)}{z}=\lim_{z\to 0}\frac{f(z)-f(0)}{z-0}=f'(0)=g(0). \end{align} Hence, $g$ is clearly holomorphic on $0<|z|<1$, and it is continuous at $0$. Under this situation, it actually follows that $g$ is also holomorphic at the origin (for example, using Morera's theorem, or by Cauchy's inequalities applied to the Laurent expansion of $g$). Now can you continue?