[Math] Complex Analysis: Difference between a disk and a circle

Question

$f$ is said to have the mean value property for circles if

$$f(z_0) = \int_{0}^{2\pi} f(z_0+re^{i\theta}) \frac{d\theta}{2\pi}$$

We are then asked to prove this for disks centered at $z_0$. I struggle to see the difference.

Answer 1

A circle is the boundary of a disk. The MVP for disks would involve integrating over the disk. It is true, and not difficult, that if $f$ has the MVP for all circles (and is continuous, let's say) then it also has the MVP for disks, but this does require proof.