I'm trying to understand the following inequality. Let $f$ be holomorphic, such that $\mathrm{Im}f(z)\geq 0$ when $\mathrm{Im}(z)>0$. Why is it that $ \displaystyle\frac{|f(z)-f(z_0)|}{|f(z)-\overline{f(z_0)}|}\leq\frac{|z-z_0|}{|z-\bar{z}_0|}? $
Since $f$ maps the upper half plane to itself, I was thinking of mapping the plane to the unit disk by some linear fractional, and then attempt to use Schwarz' lemma somehow. I haven't been able to execute a good plan.
Does anyone have any hints and/or solutions to show this inequality? Thank you.