A comp exam problem asks us to prove "the following piece of the Riemann Mapping Theorem."
If $f,g$ are analytic bijections from an open set $A$ to the unit disc $D$ with $f(a) = g(a) = 0$ and $f'(a),g'(a)>0$ for some $a\in A,$ show that $f=g$ on $A$.
Unless there is some reason that $f'(a),g'(a)$ must be real numbers, it doesn't make sense to write $f'(a),g'(a)>0,$ so what is the correct statement? Beyond that, I have no idea how to begin to prove this. Any hints or suggestions are appreciated.
$\Bbb C$ is not an ordered field, but $w > 0$ is a frequently used short form for "$w$ is a positive real number".
You can prove the uniqueness of the Riemann mapping by applying the Schwarz lemma to both $h = g \circ f^{-1}$ and $h^{-1}$.