Schwarzian derivatives and uniformization

In every textbook on elementary function theory you can find a definition of Schwarzian derivative {z; x} = 1 2 z z − 1 4 z z 2 , where = d/dx, of a non-constant function z = z(x) with respect to x. You would also find an exercise to show (0.1) [PGL(2)-invariance] If a, b, c, and d are constants satisfying ad − bc = 0, then az + b cz + d ; x = {z; x}. (0.2) {z; x} = 0 if and only if z(x) = (ax + b)/(cx + d) for some constants a, b, c, and d satisfying ad − bc = 0. (0.3) [change of variable] If y is a non-constant function of x, then {z; y} = {z; x} dx dy 2 + {x; y}. (0.4) [local behavior] If z = x α u (α = 0), where u is a holomorphic function of x non-vanishing at 0, then {z; x} = 1 − α 2 4x 2 + a function holomorphic at 0 x ; if z = log(xu), where u is as above, then {z; x} = 1 4x 2 + a function holomorphic at 0 x. In this paper we discuss various generalizations of the Schwarzian derivative. We start from recalling how it was found.