Möbius transformations from the real axis to the unit circle ($w=\lambda (z+k)/(z+\overline{k})$)

A nonconstant Möbius transformation mapping the real line into the unit circle has the displayed form with $|\lambda |=1$ and $k$ nonreal. Indeed $|ax+b{|}^2=|cx+d{|}^2$ for real $x$ yields $|a|=|c|\ne 0$, $|b/a|=|d/c|$ and $\mathrm{Re}(b/a)=\mathrm{Re}(d/c)$. Thus $d/c$ is $b/a$ or its complex conjugate; the first possibility makes the map constant, so the second is necessary. Conjugate numerator and denominator have equal modulus on the real line, proving the converse as well.