Obtaining a given bound for an analytic function

Question

Let $f:D(0,1) \rightarrow \mathbb{C}$ be analytic, and assume that $Re(f(z)) > 0$ for all $z \in D(0,1)$ and $f(0)=1.$ Prove that for all $z \in D(0,1),$ $$|f(z)| \leq \frac{1+|z}{1-|z|}.$$

My approach: I considered the function $g(z)=\frac{1-z}{1+z} f(z)$ and tried to show that it's bounded by $1$ using the given information. But I wasn't successful. Any help is much appreciated.

Answer 1

Hint: You should compose your map $w=f(z)$ with $h:w\mapsto \frac{w-1}{w+1}$. Show that $h\circ f$ maps $D$ to $D$ and $0$ to $0$ and use a standard result (Schwarz Lemma) for such a map...