Let $f$ be a holomorphic function on D = $ ( z\in C : |z| <1 ) $ such that $ | f(z)|\leq1$. Let $ g : D: \rightarrow C $ be such that
$ g(z) = \frac{ f(z)} {z} $ if $z\in D $, $ z\neq 0$ and $ g(0) = \ f' (0) $ .
I have to select which are the correct options.
1) g is holomorphic (Seems correct by definition)
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$.
The solution set says all four are correct. Please suggest.