Problem with a Moebius transformation from disk to itself

Question

I need to show that a) every Möbius transformation that maps the disk onto itself can be written as $g(z)=e^{i\theta}\frac{z-z_0}{\bar{z_0}z-1}$, and b) that this function maps the unit disk onto the unit disk. I'm currently trying to make sense of the last part. Taking modules I get the following result: $|z|^2+|z_0|^2<1+|z||z_0|$, and I have been stuck there since. How can I prove this inequality given that $|z|,|z_0|<1$?

Answer 1

$$|g(z)|^2=\frac{|z-z_0|^2}{|\overline{z_0}z-1|^2} =\frac{|z|^2+|z_0|^2-(z\overline{z_0}+\overline{z}z_0)} {|zz_0|^2+1-(z\overline{z_0}+\overline{z}z_0)}{} $$ Both numerator and denominator are positive, and denominator minus numerator is $$1+|zz_0|^2-|z|^2-|z_0|^2=(1-|z^2|)(1-|z_0^2|)>0$$ whenever $z$ and $z_0$ are in the open unit disc.