To show that a complex-valued function is injective Given a complex-valued function $w=f(z) = - \dfrac{1}{2} \left( z + \dfrac{1}{z} \right)$ from $\{ z= x+iy : |z| < 1 \}$ to $\{ w : \text{Im}(w) >0 \}$, show that $f$ is injective.
My Approach : Let $f(z_1) = f(z_2)$ which yields
$$(z_1-z_2) + \left( \frac{\overline{z_1}}{|z_1|^2} - \frac{\overline{z_2}}{|z_2|^2} \right) = 0 .$$
From here how can I conclude that $z_1 = z_2$ ? Any help is much appreciated.

Alt. hint: $\;f(z)=w \iff z^2 + 2w z+ 1 = 0\,$. The quadratic has two roots whose product is $1$, so only one of them can be inside the unit circle $|z| \lt 1$.
