Schwarz Lemma/Conformal mapping problem

Question

Let $F:\mathbb{H}\rightarrow \mathbb{D}$ be holomorphic, where $\mathbb{H}$ is the upper half plane and $\mathbb{D}$ is the unit disc. Show that if $F(i)=0$, then $$|F(z)|\leq \left|\frac{z-i}{z+i}\right|$$ for all $z\in\mathbb{H}$.

I have trouble constructing an auxiliary function to apply the Schwarz Lemma or something, any hints?

Thanks

Answer 1

Well, the answer is in your question.

The conformal mapping from the unit disk to the upper half-plane is $G=\frac{z-1}{z+1}$. So, $F\circ G$ will satisfy the hypotheses of the Schwarz Lemma.

If you want to prove that this is your desired function, I suggest the chapter on conformal mappings in Bak/Newman.