CWMO 2015 Problem 7

Let $a\in (0,1)$ , $f(z)=z^2-z+a, z\in \mathbb{C}$ . Prove the following statement holds: For any complex number z with $|z| \geq 1$ , there exists a complex number $z_0$ with $|z_0|=1$ , such that $|f(z_0)| \leq |f(z)|$ .