Question: Let $f$ be analytic and Im$f(z)>0$. Show $|\frac{f(z)-f(z_0)}{f(z)-\overline{f(z_0)}}|\leq|\frac{z-z_0}{z-\overline{z_0}}|$.
Thoughts: A nearly identical problem is asked here: Why does $\frac{|f(z)-f(z_0)|}{|f(z)-\overline{f(z_0)}|}\leq\frac{|z-z_0|}{|z-\bar{z}_0|}$ when $\mathrm{Im}z>0\implies\mathrm{Im}f(z)\geq 0$?, but it is showing the other direction. I felt like I could prove it in a similar fashion, but I don't have the "Im$(z)\geq 0$" assumption, just that Im$f(z)>0$, unless I am not seeing how that still holds then in this case. I thought about maybe using some Schwarz here, but I thought there might be a way to salvage the proof from the attached problem. Any help is greatly appreciated! Thank you.