Prove ff is unique

Let AA be a simply connected region in C\mathbb{C}

Let Δ={z:|z|<1}\Delta=\{z : |z|\lt 1 \} Prove that if ff is a conformal and bijective map, so that it sends a point z0∈Az_0 \in A to the origin and f′(z0)>0f'(z_0 )\gt 0 , then ff is unique.

Any hints on how to prove this? I’m new to complex analyisis (using Marsden’s Book) and I’m a bit lost.



1 Answer


Hint. Let g:A→Δg:A\to \Delta be another function which satisfies this property.

h:=g∘f−1:Δ→Δh:=g\circ f^{-1}:\Delta\to \Delta is a conformal and bijective map such that h(0)=0h(0)=0 and h′(0)=g′(z0)/f′(z0)>0h'(0)=g'(z_0)/f'(z_0)>0.
Then use Schwarz Lemma.