Average value of $\log |z_0 - z|$

Question

I am interested in the average value of $\log |z_0 - z|$ as $z$ ranges over the unit circle.

By the mean value property for harmonic functions, I understand that when $z_0$ lies outside this circle, i.e. $|z_0| > 1$, then the value of this average is simply $\log z_0$.

How do I calculate the average in the other case, where $|z_0| \le 1$?

Answer 1

Hint: Suppose $|z_0|<1.$ Then

$$|z-z_0| = |1-\overline{z_0}z |$$