I try to solve the following exercise from "Function of One Complex Variable I, Conway"
Let $p(z)$ be a non-constant polynimial and $c$ be a positive real number. Then each component of $|p|^{-1}(<c) = \{z \in \mathbb{C} : |p(z)| < c\}$ contains a zero of $p$. (Chapter 6)
Hint : Use the following exercise
Let $G$ be a bounded region and suppose $f$ is continuous on $\overline{G}$ and holomorphic on $G$. Show that if there is a constant $a \geq 0$ such that $|f(z)| = a$ for all $z$ on the boundary of $G$, then $f$ is constant or $f$ has a zero in $G.$ (I can prove the hint)
$\textbf{Proof}$ Let $A$ be a component of $|p|^{-1}(<c)$. Since $p$ is non-constant, if $|p|$ is constant on the boundary of $A$, then the hint give the result.
I just not sure how to deduce that $|p|$ must be constant on the boundary of $A.$
This follows from continuity of $|p|.$
Namely, if $A$ is a component of $|p|^{-1}(<c)$ and $\partial A$ is the boundary, then you have that each point $a \in \partial A$ is a limit of points $a_n \in A.$ You know that $|p(a_n)| < c,$ so by continuity, $|p(a)| \leq c.$ You can not have $|p(a)| < c,$ since then we would have that $a \in A.$ Now you can conclude with the hint.