Automorphisms of the unit disc is less than 1

[Likemath2014]

I want to show that the modulus of the automorphism

$$\frac{a-z}{1-\overline{a}z}$$

is strictly bounded by 1 in the unit disc. Applying Schwarz lemma gives the result immediately. But I am looking for a straight forward proof for that.

Thanks in advance

 

[jbunniii]

So you need to prove that $|a-z| \leq |1 - \overline{a}z|$ whenever $|z| \leq 1$. Equivalently, you require
$$(a-z)(\overline{a}-\overline{z}) \leq (1 - \overline{a}z)(1 - a\overline{z})$$
Performing the multiplication on both sides, we need
$$|a|^2 - 2\text{Re}(\overline{a}z) + |z|^2 \leq 1 - 2\text{Re}(\overline{a}z) + |a|^2|z|^2$$
It should be straightforward from this point.

 

[Likemath2014]

Now it is clear, and the last one is true because
$$(1-|z|^2)(1-|a|^2)>0.$$
Thanks