Let $$\varphi:\mathbb{H}\rightarrow\mathbb{D},~z\mapsto\frac{z-i}{z+i}$$ be the biholomorphic map from the upper half-plane $\mathbb{H}$ to the open unit disk $\mathbb{D}$.
I wonder how the image of this set under $\varphi$ looks like.
Is this possible with Mathematica?
Here are some images. Made with Maple (my code is not very pretty though).
All the data here are approximate. The green circle is centered on $1+2i$ and of radius $2$. The yellow circle is then centered on $1-2i$ and of radius $2$. The most high red circle is centered on $-0.71+0.71i$ and of radius $0.14$. The most low then on $-0.71-0.71i$ with the same radius. Finally the two others are centered on $-1+0.333i$ and $-1-0.333i$ with the same radius $0.333$.