inequality of complex numbers

Question

How to prove $$\left|\frac{a-b}{1-\bar{a}b}\right|<1,$$

where $a,b$ are complex numbers, $\bar{a}$ is the complex conjugate of $a$, and $|a|,|b|<1$.

Answer 1

$$|a-b|^{2} -|1-\overline {a} b|^{2}=[|a|^{2}+|b|^{2}-2Re (\overline {a} b)]-[1+|a|^{2}|b|^{2} -2Re (\overline {a} b)]$$ $$=-(1-|a|^{2})(1-|b|^{2}) <0$$.