On the Harmonic Möbius Transformations

It is well-known that two locally univalent analytic functions have equal Schwarzian derivative if and only if each one of them is a composition of the other with a non-constant Möbius transformation. The main goal in this paper is to extend this result to the cases when the functions considered are harmonic. That is, we identify completely the transformations that preserve the (harmonic) Schwarzian derivative of locally univalent harmonic functions.