the Schwarzian derivative
/ SHVART-see-un /
Most derivatives measure how a function changes. The Schwarzian derivative measures something more exotic: how much a map deviates from being a Mobius transformation — a fractional-linear map like (az + b)/(cz + d). It is built to be blind to exactly those maps, vanishing precisely on them, which makes it the natural gauge of 'higher-order bending' and a surprisingly sharp tool for deciding when a holomorphic function is univalent.
Precisely, for a function f with f' nonzero, the Schwarzian derivative is Sf = (f''/f')' - (1/2)(f''/f')^2, often written {f, z} = f'''/f' - (3/2)(f''/f')^2. Two key facts: (1) Sf = 0 if and only if f is a Mobius transformation, so the Schwarzian truly measures departure from Mobius. (2) It satisfies an elegant composition (cocycle) rule, and is invariant when you post-compose f with a Mobius map — S(M circle f) = Sf for Mobius M. That invariance is exactly why it appears in problems where the Mobius freedom should be quotiented out, such as univalence criteria and second-order linear ODEs (the Schwarzian connects to the equation u'' + (1/2)(Sf) u = 0).
Its starring role here is Nehari's univalence criterion: if f is holomorphic on the unit disk and its Schwarzian is not too large — specifically |Sf(z)| <= 2/(1 - |z|^2)^2 for all z in the disk — then f is univalent on the disk. The bound 2 is sharp. So a single inequality on the Schwarzian certifies global one-to-one-ness, a remarkable shortcut past directly verifying injectivity. A caution: the criterion is sufficient, not necessary (a univalent map can violate it), and the conjugate-friendly constant is delicate — a slightly different constant (Nehari also gave a version with 2/(1 - |z|^2)^2 replaced by a pi^2-type bound) is needed for related sharp statements. Match the constant to the exact theorem you are quoting.
Compute the Schwarzian of any Mobius map, say f(z) = (z + 1)/(z + 2): here f'' and the combination cancel exactly, giving Sf = 0 — confirming that the Schwarzian vanishes precisely on fractional-linear maps. For f(z) = z^2, f''/f' = 1/z, so Sf = -1/z^2 - (1/2)(1/z^2) = -3/(2 z^2), nonzero, as expected for a non-Mobius map.
Sf = 0 exactly for Mobius maps; Nehari: |Sf| <= 2/(1 - |z|^2)^2 forces univalence.
Nehari's bound is sufficient for univalence, not necessary, and the sharp constant depends on which version you cite. The Schwarzian is invariant under post-composition by Mobius maps but NOT under pre-composition.