Find a set of complex numbers for equation $2|z|<|1+z^2|$ and draw it

Question

Find a set of complex numbers for equation $2|z|<|1+z^2|$ and draw it.

That's as far as I got: $$4x^2+4y^2<(x^2+y^2+1)^2+x^2y^2\\ \vdots\\ x^4+y^4-6x^2-2y^2+1>0\\ (x^2-3)^2+(y^2-1)^2>9$$

Now I have no idea how to draw this.

Answer 1

$2\vert z \vert <\,\vert 1+z^2\vert \,<\,1+\vert z \vert ^2 $ $$ \vert z \vert ^2 -2\vert z \vert +1>0$$ $$(\vert z \vert -1)^2>0$$ $$\vert z \vert \ge 0 \,\,and \vert z \vert \neq 1 $$