Defining a new function to be holomorphic by construction

Question

Given that some $f(z)$ is analytic on the unit disc, $f(0) = 0$, and $|f(z)| \leq M$, I want to show that $|f(z)| \leq M|z|$ on the unit disc.

I want to use the Maximum Modulus Principle (MMP) on a new function, $h(z)$, which will be defined as $\frac{f(z)}{z}$ everywhere but $0$, and be defined at $0$ such that $h(z)$ is holomorphic. Given that $f(0) = 0$, I was thinking of defining $h(0) = 0$, but am not sure how to then prove that $h(z)$ is holomorphic at $0$.

I tried the limit approach, but wasn't really sure how to proceed beyond just writing it down.

Thanks for your help!

Answer 1

Hint: The derivative is defined by $$f'(z)=\lim_{h\to 0} \frac{f(z+h)-f(z)}{h}$$

Alternatively you could argue with the power series of $f$.