Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$? Is there anything special with the form:
$$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$
? With $a$ and $z$ are complex numbers.
In fact, I saw it in a problem:


*

*If $|z| = 1$, prove that $|\frac{z-a}{1-\bar{a}{z}}| = 1$

*If $|z| < 1$ and $|a| < 1$, prove that $|\frac{z-a}{1-\bar{a}{z}}| < 1$


I can easily prove the first one with expansion:
$$z=\cos\theta + i\sin \theta \\ a = m +in$$
But it will be terrible to use in the second one. What's more, I found this form a little special so maybe there is some clever trick without using expansion?
Thank you a lot!

Hint: Use $z\overline{z}=|z|^2$ and compare $|z-a|^2$ to $|1-\overline{a}z|^2$.
Have a look at the comments to see why these transformations are special.
