# 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:

1. If $|z| = 1$, prove that $|\frac{z-a}{1-\bar{a}{z}}| = 1$
2. 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!

#### Solutions Collecting From Web of "Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?"

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.

We first prove that if $|z|=1$ then $|f(z)|=1$. For we have
$$|f(z)|=|\frac{z-a}{1-\bar az}|=|\frac 1{\bar z}\frac{z-a}{1-\bar az}|=|\frac {z-a}{\bar z-\bar a}|=1$$
Now apply maximum principle to get $|f(z)|<1$ for $|z|<1$.