We study the Schwarzian derivative from a variational viewpoint. Firstly we show that defines first integral of Euler--Lagrange equation second order Lagrangian. Secondly, itself is operator for an appropriately chosen class variations.

Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-frac...

Let f be a locally injective analytic map of the unit disc D and let ff; zg be its Schwarzian derivative. Suppose jff; zgj 2p(jzj). We use the classical connection between Schwarzian derivative and second order linear equations to show that, for a particular class of functions p , the image f (D) is a quasidisc. The analysis centers on the diierential equation y 00 + py = 0 and a niteness condi...

The flag curvature of the Numata Finsler structures is shown to admit a nontrivial prolongation to the one-dimensional case, revealing an unexpected link with the Schwarzian derivative of the diffeomorphisms associated with these Finsler structures. Mathematics Subject Classification 2000: 58B20, 53A55 1 Finsler structures in a nutshell 1.1 Finsler metrics A Finsler structure is a pair (M,F ) w...

We provide two new formulas for quasiconformal extension to C‾ harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of Ahlfors-Weill holomorphic functions.