Strangely, I don't find easily on the internet sources about inequalities with complex numbers. In this moment, I am interested to absolute value inequalities with complex numbers but would be good having a wider perspective with all sorts of inequalities with complex numbers.
I have here an exercise which asks: Describe the set $C=\{z: |z−2i|≤ 2\} $ as a subset of the complex plane. Draw a picture.
It doesn't look hard but I have some questions. Usually, am I wrong if I say that we operate this way in R:
$-2 <= |z−2i|≤ 2$ then, we should isolate the variable within the absolute value - let's say that "-2" within the absolute value is a real constant - and we would end up with: $0 <= |x| <= 4$ which would be the interval solution for our inequality. This is because the absolute value theory in R states that we calculate the distance from the origin of the number line.
Anyhow, we are in C now. As I have recently learnt, we consider the magnitude of the complex number seen as a vector - which is quite intuitive and thus easy to learn. However, I don't see in this case the real part of the complex number - am I wrong if I say that we are dealing only with the imaginary part? As z ∈ C, so I am still not sure whether there is a real part or not...
Since this is the first time that I try to solve an inequality with a complex number and I don't find enough material on the internet, I ask here. This is my guess, however, based on logical thinking about inequalities: do I have consider as interval solution of the inequality all the values which don't make the magnitude of the vector zero or negative?
To Henry's comment here is the drawing: