Let \(D^+\coloneqq\theset{z\in\CC : \abs{z}<1,\ \Im z>0}\) be the upper half-disc. Then \[ \begin{aligned} F\colon D^+ &\to \HH, \\ z & \mapsto -{1\over 2}\qty{z + \inverseof{z}} \end{aligned} \] is a biholomorphism.
For \(z=re^{i\theta}\) with \(0<r<1\) and \(0<\theta<\pi\), \(\Im F(z)=-\frac12\qty{r-\inverseof{r}}\sin\theta>0\), so \(F(D^+)\subseteq\HH\). For \(c\in\HH\), the solutions of \(F(z)=c\) are the roots of \(z^2+2cz+1=0\), whose product is \(1\). A root on the unit circle would make \(F(z)=-\Re z\) real, so neither root lies on the circle and exactly one root \(z\) satisfies \(\abs{z}<1\); then \(\Im F(z)>0\) forces \(\sin\theta>0\), so \(z\in D^+\). Hence \(F\colon D^+\to\HH\) is a holomorphic bijection, and a bijective holomorphic map has a holomorphic inverse.