Exercise 1, § 2

Let $a$ be a non-zero complex number, and $n$ an integer > 0. For each real number $r > 0$ such that $r^n \leq |a|$, show that there exists $z \in \mathbf{C}$ such that $|z| = r$ and $|a + z^n| = |a| - r^n$. Deduce that, if $f$ is a polynomial of degree > 0, with complex coefficients, we cannot have $|f(z_0)| \leq |f(z)|$ at all points $z$ of a neighbourhood of a point $z_0$, provided that $f(z_0) \neq 0$.