If $f$ is a holomorphic function from the upper half-plane(without the real line) to the unit disc, and if $f(i) = 0$, how large can $|f(2i)|$ be under these conditions?
I'm not sure what theorems and techniques to use here. Schwarz's lemma or cauchy estimates? But I'm not sure how to use them here. Any suggestions? This is from a Dutch complex analysis textbook
Best Answer
The upper half plane is conformally equivalent to the unit disc by the Cayley transform $$\varphi:\mathbb H\longrightarrow\mathbb D,~z\mapsto\frac{z-\mathrm i}{z+\mathrm i}.$$ The Cayley transform also maps $\mathrm i\mapsto0$, so its inverse $\varphi^{-1}$ maps $0\mapsto\mathrm i$.
Now $f\circ\varphi^{-1}$ is a holomorphic map $\mathbb D\longrightarrow\mathbb D$ which maps $0$ to $f(\varphi^{-1}(0))=f(\mathrm i)=0$. Now apply the Schwarz lemma to $f\circ\varphi^{-1}(\varphi(2\mathrm i))=f\circ\varphi^{-1}(\frac{1}{3})$.
It is generally a useful trick to consider conformal maps to the unit disc, because the Schwarz lemma tells us a lot about how holomorphic functions behave on the disc. So if we have a holomorphic function $f:A\longrightarrow B$, where $A$ and $B$ are conformally equivalent to the unit disc via the conformal maps $\varphi_{A}:A\longrightarrow \mathbb D$ and $\varphi_{B}:B\longrightarrow \mathbb D$, then the function $\varphi_B\circ f\circ\varphi_A^{-1}$ is a holomorphic map from the unit disc to the unit disc. If we also manage to choose $\varphi_A$ and $\varphi_B$ such that this map $\varphi_B\circ f\circ\varphi_A^{-1}$ maps $0$ to $0$, then we can apply the Schwarz lemma to the composite function, which can yield desirable insights.