Consider $f:\mathbb{D}\to \mathbb{D}$ (where $\mathbb{D}$ is the unit disc), which moves two points $z_1, z_2\in \mathbb{D}$ to $w_1, w_2\in \mathbb{D}$. I want to prove that if such an $f$ exists, then it satisfies $$\left| \frac{w_1-w_2}{1-\bar{w_2}w_1}\right|\leq \left| \frac{z_1-z_2}{1-\bar{z_2}z_1}\right|.$$
There is the hint saying that one might try moving both $z_2$ and $w_2$ to $0$ first. But I have no idea why $w_2$ would need to be moved to $0$ at all, because $z_1, z_2$ need to be moved to $w_1, w_2$. Why move $w_2$ to $0$ then?
Anyway, we can define $\rho_{w_2}(z) = \frac{w_2-z}{1-\bar{w_2}z}$ and $\rho_{z_2}(z) = \frac{z_2-z}{1-\bar{z_2}z}$. Then we can consider $\rho_{w_2}(w_1)$ and $\rho_{z_2}(z_1)$. I don't know, though, what these manipulations would be for, except for just trying to derive the parts of the inequality. But then, where does the inequality come from?
Conversely, if $$\left| \frac{w_1-w_2}{1-\bar{w_2}w_1}\right|\leq \left| \frac{z_1-z_2}{1-\bar{z_2}z_1}\right|,$$ then how can we find the function $f:\mathbb{D}\to\mathbb{D}$ that takes $z_1, z_2$ to $w_1, w_2$?
Unfortunately, I don't know much about interpolating functions.
I'd really appreciate some hints.