When $f$ holomorphic, $|f(z)-f(0)-f'(0)z|\leq 3c|z|^2$

Question

$f$ be a holomorphic function on $A:=\{z; |z|\leq 1\}$, and $c:=\max {|f(z)|}$. Then, for all $z\in A$, $|f(z)-f(0)-f'(0)z|\leq 3c|z|^2$

By using Laurent expansion and Cauchy's formula I can prove if $|z|$ small, but I can't prove in general.

Answer 1

Let $$g(z)=\frac{f(z)-f(0)-f'(0)z}{z^2}.$$ Then $g$ is holomorphic on $A$. When $|z|=1$, $$|g(z)|\le c+|f(0)|+|f'(0)|\tag{*}$$ and so by the maximum modulus theorem (*) holds on $A$. It's clear that $|f(0)|\le c$, so to complete the proof, all you need to prove is that $|f'(0)|\le c$.