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 ...

S f t,,-,] \-fT,I f 2 t,,--fT,] The Schwarzian is important in many areas of complex analysis (see, for example, the recent book of O. Lehto ILl) but it occurs first and foremost through its connection with M6bius transformations. The basic facts are az+b (1.1) S(f) 0 if and only iff(z) ad bc :/: 0 cz+d’ and (1.2) S(fo h) S(h) if and only iff is M6bius. Equation (1.2) is a special case of a gen...

Theorems due to Nehari and Ahlfors give sufficient conditions for the univalence of an analytic function in relation to the growth of its Schwarzian derivative. Nehari's theorem is for the unit disc and was generalized by Ahifors to any simply-connected domain bounded by a quasiconformal circle. In both cases the growth is measured in terms of the hyperbolic metric of the domain. In this paper ...

We begin by considering the set of infinitely renormalizable unimodal maps on the interval [−1, 1]. A function f defined on [−1, 1] is said to be unimodal if it is continuous, increasing on [−1, 0], decreasing on [0, 1] and symmetric about 0, and if it fixes −1 and maps 1 to −1 . Moreover, it is said to be renormalizable if there is an integer n > 1 and a subinterval I 6= [−1, 1] containing 0 s...

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, ...