In a textbook Im reading they write the following:
Let $h,\hat h:D \rightarrow G\subset \mathbb{D}$ (unit disk) two conformal functions. If there is a point $a \in G$ with $h(a) = \hat h(a)$ and $h'(a) / \hat h'(a) > 0$ then follows $h = \hat h$.
Im concerned about $h'(a) / \hat h'(a) > 0$
$h'(a) / \hat h'(a)$ should be complex valued and one cant say $h'(a) / \hat h'(a) > 0$. I think what they want to say is $h'(a) / \hat h'(a) \neq 0$.
Or is the derivative of a conformal function always real? This does not make sense to me.
Generally, when people writes (in the context of complex valued function)
$f(a)>0$ for some $a$ in the domain
it means $f(a) \in \mathbb R$ and $f(a)>0$.
(cf. Uniqueness part of Riemann mapping theorem)
Derivative of conformal function need not be real. $h'(a) / \hat h'(a) > 0$ is just a part of the hypothesis of your statement.