Plot $|\frac{z+i}{z-1}|<1$ Let $D=\{|\frac{z+i}{z-1}|<1\}$ plot $D$
$$\frac{z+i}{z-1}|<1\iff \frac{|z+i|}{|z-1|}<1\iff |z+i|<|z-1|\iff |x+(y+1)i|<|(x-1)+yi|\iff\\ \iff \sqrt{x^2+(y+1)^2}<\sqrt{(x-1)^2+y^2}\iff {x^2+(y+1)^2}<{(x-1)^2+y^2}\iff \\ \iff x^2+y^2+2y+1<x^2-2x+1+y^2\iff x<-y$$
Which is the are under the line $x=-y$ 
Is it correct?

$$\frac{|z+i|}{|z-1|}<1$$
$$|z-(-i)| < |z-1|$$
We are describing points that are closer to $(-i)$ than $1$.
The perpendicular bisector between the two points is $y=-x$ and the corresponding region is the area under $y=-x$.
