$G\subset \mathbb{C}$ simply connected, $f$ holom. on $G$, show every connected comp. of $\{ |f| < c\}$ $(c > 0)$ is simply connected.

Question

I have the tools:

1. $G$ simply connected $\iff$ every contour in $G$ has winding number $0$ about every point in $\mathbb{C}\setminus G$

2. $G$ is simply connected $\iff$ every holom. function on $G$ has a primitive.

Ideas: I could look at an arbitrary connected component of $S_{c} := \{|f| < c\}$ and use the primitive idea to say something about winding number? I tried to think about this and write some stuff down but I have no luck.

I tried showing (1) directly without any ideas from (2) but also with no luck. I even tried using the fact that $G$ is simply connected - I can write $G = S_{c} \cup G\setminus S_{c}$ and view $f$ on each set, but again I don't know where to go from here.

I am pretty lost on this problem. I would like a hint, not a solution. Thank you!

Answer 1

There is another way to show a region is simply connected:

$G$ is simply connected $\iff$ The interior of any simple closed curve $\gamma\subseteq G$ is contained in $G$.

Now take a simple closed curve $\gamma\subseteq \{|f|<c\}$. The interior $U$ of $\gamma$ is contained in $G$ since $G$ is simply connected. Then apply the maximum modulus principle.