properties of complex modulus question If $|a| < 1$, prove that $|z| < 1$ is equivalent to $$\frac{|z - a|}{|1-\bar{a}z|} \leq 1.$$ Where $a$ and $z$ are complex and $\bar{a}$ denotes the conjugate of $a$. I thought this was the easiest problem I would have to do tonight, but I have been bumbling around with the properties of the modulus ($|z + a| \leq |z| + |a|$, etc.) all night and I can't get the desired result. Is this a bad question?

**Hint**. First rewrite the inequality as $$|z-a|\le|1-\overline a z|\ ,$$ then use the relation $$|w|^2=w\overline w\ .$$ This is often a good way to go because conjugates have "nicer" algebraic properties than magnitudes. Squaring the left hand side, $$|z-a|^2=(z-a)\overline{(z-a)}=(z-a)(\overline z-\overline a)=z\overline z-a\overline z-z\overline a+a\overline a\ .$$ Do something similar for the right hand side and see if you can take it from there.

Also don't forget that the question said "equivalent".