Inequality for modulus Let $a$ and $b$ be complex numbers with modulus $< 1$.
How can I prove that 
$\left | \frac{a-b}{1-\bar{a}b} \right |<1$ ?
Thank you

Here are some hints: Calculate $|a-b|^2$ and $|1-\overline{a}b|^2$ using the formula $|z|^2=z\overline{z}$. To show that $\displaystyle\left | \frac{a-b}{1-\bar{a}b} \right |<1$, it's equivalent to show that 
$$\tag{1}|1-\overline{a}b|^2-|a-b|^2>0.$$
To show $(1)$, you need to use the fact that $|a|<1$ and $|b|<1$. 
If you need more help, I can give your more details.
