Just want to check. What is the limit of function $\frac{z}{\bar{z}-z}$ at $z=0$? I got $\lim_{\substack{z \to 0 \\ z \in \mathbb{R}}} \frac{z}{\overline{z}-z} =-\infty$
and $\lim_{\substack{z \to 0 \\ z \in i\mathbb{R}}} \frac{z}{\overline{z}-z} =-\frac{1}{2}$, so $f$ is not defined at $z=0$? Byt the way does this have any singularities? And finally is this analytic in unit circle?