Möbius maps commuting with reflection in the unit circle
= Möbius maps commuting with reflection in the unit circle
{c}
The Möbius transformations commuting with $J(z)=1/\overline z$ are exactly
$$
z\longmapsto\frac{az+b}{\overline bz+\overline a},
\qquad |a|^2-|b|^2\ne0.
$$
They preserve the open unit disc exactly when $|a|>|b|$. The identity
$$
|az+b|^2-|\overline bz+\overline a|^2
=(|a|^2-|b|^2)(|z|^2-1)
$$
proves the latter assertion.