Let $D$ denote the unit disc (|z| < 1). Let $a \in \mathbb{C}, B \in \mathbb{R}$ and $r > 0.$ I want to find a Mobius map $f$ such that
(1) $f(D) = \{z = x+iy : |z-a| < r\}$
$\textbf{Sol}$ Let $g(z) = rz$ and $h(z) = z-a$. Then $f = h \circ g.$
(2) $f(D) = \{z \in \hat{\mathbb{C}} : |z-a| > r\}$
$\textbf{Sol}$ Let $g(z) = rz$, $h(z) = z-a$ and $k(z) = z^{-1}.$ Then $f = k \circ h \circ g.$
(3) $f(D) = \{z = x+iy : x > B\}$
$\textbf{Sol}$ Let $g(z) = i\frac{1-z}{1+z}, h(z) = e^{i(-\pi/2)}z$ and $k(z) = z+B.$ Then $f = k \circ h \circ g.$
Is it correct ?
$z \mapsto r z$ scales the unit disk by the factor $r$, then $z \mapsto z - a$ translates the disk so that the center moves to $-a$. You want $(z \mapsto z + a) \circ (z \mapsto r z)$ instead.
$z \mapsto 1/z$ performs the combination of plane inversion with conjugation. The center and the radius of the disk will change if $z \mapsto 1/z$ is applied as the third step. You want $z \mapsto r/z + a$.
The third mapping is correct, $g$ maps the unit disk to the upper half-plane.