Proving $\left|\frac{w-z}{1-\bar{w}z}\right|$ < 1

Question

How to prove $\left|\frac{w-z}{1-\bar{w}z}\right|$ < 1 if |z|<1 and |w|<1?

Please give me a hint.

Answer 1

Hint:

$$\left|\frac{w-z}{1-\overline wz}\right|<1\iff|w-z|^2<|1-\overline wz|^2\iff$$

$$\iff (w-z)(\overline w-\overline z)<(1-\overline wz)(1-w\overline z)\iff$$

$$\iff |w|^2+|z|^2-2\,\text{Re}\,(w\overline z)<1-2\,\text{Re}\,(w\overline z)+|w|^2|z|^2\iff$$

$$\ldots\text{last purely algebraic step yielding an obvious inequality...}$$