I'm looking for information on sets of the form $|f(z)| = c$, possibly with additional restrictions (e.g. $|z| < d$). I can probably also assume that $f$ is as nice as you'd like.
I'm interested in things like:
Into how many regions does the curve split the plane?
When is $|f(z)| = c$ path connected?
When is $|f(z)| \leq c$ bounded?
When is $|f(z)| \leq c$ simply connected?
If $(f_n)$ is a sequence of functions, what is the limiting behavior of $|f_n(z)| = c$?
If you could point me toward any references I'd be very grateful. I'd also be interested in references where specific cases are treated, just so I can get an idea of the common techniques.
I know this question is pretty broad but I'm not looking for a particular answer. Any information you could provide would be a great help.