Suppose $$\begin{align*} f: D(0,1) \to D(0,1) \end{align*}$$ is a holomorphic function on $D(0,1)$ which is the unit disk. Then I want to prove that $$\begin{align*} \frac{\left| |f(0)| -|z| \right| }{1-|f(0)||z|} \leq |f(z)| \leq \frac{|f(0)| + |z|}{1+|f(0)| |z|} \end{align*}$$ I think I need to use Schwarz lemma to prove this inequality, and I construct a function $$\begin{align*} F(z) = \frac{f(z) - f(0)}{1- \overline{f(0)}f(z)} \end{align*}$$ and I think $|F(z)| <1$ and $F(0)= 0$ . Thus $F$ is a holomorphic function from unit disk to unit disk. Then by Schwarz lemma, we have $$\begin{align*} |F(z) |&\leq |z| \\ f(z) &= \frac{F(z) + f(0)}{1+ \overline{f(0)}F(z)} \\ |f(z)| & \leq \frac{|F(z)| +|f(0)|}{1-|f(0)| |F(z)|} \\ & \leq \frac{|z| +|f(0)|}{1-|f(0)| |z|} \end{align*}$$ However, by using the triangle inequality, I can only obtain $1-|f(0)||z|$ at the denominator, not $1+|f(0)||z|$. Thus, I'm wondering if probably I need to use other inequalities or if my construction of $F$ is not suitable. Any help? Thanks.
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