I'm trying to sketch $D=\{z \in \mathbb{C}: 2 \leq \vert z \vert < \vert z-2 \vert <4\}$.
I know that, geometrically, this is all $z$ whose distance from the origin is $\geq 2$ and is $<$ whose distance from $2$, which is less than $4$.
The question is: how do I sketch this information in the complex plane? Here's my attempt: the red represents the unwanted region and the green the desired region. Is this correct?
Thanks
Take the inequalities one at a time. The region $2 \le |z| < 4$ is an annulus, i.e. region between two circles, in this case including the inner circle of radius 2 centered at $0$ and excluding the outer circle of radius 4 (also centered at $0$).
The region $2 < |z-2| < 4$ is another annulus, this time not including either bounding circle, but centered at $2$. Finally the requirement $|z|<|z-2|$ describes the region to the left of the vertical line $x=1$ where $z=x+yi.$
Putting these together I got something that looks like two sections of "moons" which join at one point.