Schwarzian chain rule

Statement

Let $f$ and $g$ be locally univalent holomorphic maps for which the composite $f\circ g$ is defined. Their Schwarzian derivatives satisfy

S(f\circ g) =\bigl(S(f)\circ g\bigr)(g')^2+S(g).

Postcomposition by a Möbius transformation

If $T$ is a Möbius transformation, then $S(T)=0$ , so

S(T\circ f)=S(f).

Thus the Schwarzian is unchanged by changing the projective coordinate on the target.

Coordinate changes

The factor $(g')^2$ is the transformation factor of a quadratic differential, while the added term $S(g)$ is the cocycle correction. This is the transformation mechanism in the definition of a holomorphic projective connection.

References

Zeev Nehari, Conformal Mapping, Dover, 1975. Relevant: Chapter VI.
R. C. Gunning, Lectures on Riemann Surfaces, Princeton University Press, 1966. Relevant: projective connections and the Schwarzian.