Holomorphic function $f$ on the unit disc such that $|f(z)|\leq 1.$ Let $f$ be a holomorphic function on $D=\{z\in\mathbb{C}:|z|\leq 1\}$ such that $|f(z)|\leq 1.$
$$g(z)=\begin{cases} \dfrac{f(z)}{z} &\text{ if } z\neq 0\\[8pt] f'(0) &\text{ if } z=0.\end{cases}$$
Which of the following statements are true
$1.$ $g$ is holomorphic on $D.$
$2.$ $|g(z)|\leq 1$ for all $z\in D.$
$3.$ $|f'(z)|\leq 1$ for all $z\in D.$
$4.$ $|f'(0)|\leq 1.$
By Riemann removablity theorem it is clear that $g$ is holomorphic on $D.$ I don't know how to handle other options. Please help me. Thanks a lot.

Hint: Maximum modulus principle for your function on $\mathbb{D}$ .  The answer is $g$ is holomorphic and $|f'(0)|\le 1$
