Equal Schwarzians differ by Möbius postcomposition

Statement

Let $D\subseteq\mathbb C$ be a connected domain, and let $f,g:D\to\widehat{\mathbb C}$ be locally univalent meromorphic maps. Then

S(f)=S(g)

if and only if there is a Möbius transformation $T$ such that

f=T\circ g

on $D$ .

Proof

The reverse implication follows from the Schwarzian chain rule and $S(T)=0$ . For the forward implication, choose a local inverse of $g$ . The chain rule shows that $f\circ g^{-1}$ has zero Schwarzian, so the vanishing-Schwarzian characterization makes it locally Möbius. The local Möbius transformations agree on overlaps and therefore give one $T$ on the connected domain.

References

Zeev Nehari, Conformal Mapping, Dover, 1975. Relevant: Chapter VI.