Let $f$ be a holomorphic function on $B(0,R)$, where $R>0$. Assume that there exists an $M>0$ such that $\forall z\in B(0,R): |f(z)| \le M$ and a natural number $n$ such that $ 0 = f(0)=f'(0) = ... = f^{(n)}(0).$
$1)$ Prove that $\forall z\in B(0,R) : |f(z)| \le M \left( \frac{|z|}{R} \right)^{n+1}$ with equality iff there exists an $\alpha \in \mathbb{C}$ such that $|\alpha|=1$ and $ f(z)=\alpha M \left( \frac{z}{R} \right)^{n+1}$.
$2)$ Assume that either $ |f(z_0)|=M \left(\frac{|z_0|}{R}\right)^{n+1}$ for some $z_0 \in B(0,R) \setminus \{0\}$ or $ |f^{(n+1)}(0)|=(n+1)!M / R^{n+1}$. Then $ f(z)=\alpha M \left(\frac{z}{R}\right)^{n+1}$ for an $\alpha \in \mathbb{C}$ such that $|\alpha|=1$.
Sorry for asking this problem, but I'd like to see some examples related to the Schwarz lemma. Thanks!