$f:U\rightarrow \mathbb C$ is holomorphic. Let $U=\{z\in \mathbb C : |z-a|\leq r \}$.
Deduce that $f(a)$ is the mean of the values of $f$ on the circle $|z-a|= r$
I parametrized $z=a+re^{it}$. Then $dz=ire^{it}dt$
Using Cauchy Int. Formula:
$$f(a)=\frac{1}{2\pi i} \oint f(z)/(z-a)dz=\frac{1}{2\pi i}\int_0^{2\pi} if(a+re^{it})=\frac{1}{2\pi}\int_0^{2\pi}f(a+re^{it})$$
Is this the answer they are looking for, in which case I don't undertand why this is the average value.