Holomorphic function inequality for function of the upper half plane

Question

I need help proving the following inequality

Let $H := \{z\in \mathbb C : \Im(z) > 0\} $ the upper half plane. Let $f: H \to \mathbb C$ be a holomorphic function with $\lvert f(z) \rvert \leq 1$ and $f(i) = 0$. Show that $$\lvert f(z) \rvert \leq \left\lvert \frac{z-i}{z+i} \right\rvert$$ for all $z\in H$.

I assume I need to somehow apply Schwarz' Lemma, but I have tried to somehow modify the function to do so and didn't get any result. Maybe there's a better way though. Any help appreciated!

Answer 1

Hint. The Cayley transform $$C(z):=\frac{z-i}{z+i}$$
maps the upper half-plane $H$ conformally onto the unit disk. Then consider the function $F=f\circ C^{-1}$ from the unit disc in itself. Note that $F(0)=f(i)=0$ and use the Schwarz' Lemma.