and that the Schwarzian is identically zero exactly for Möbius transformations. Let D denote the unit disk. There has been progress in several areas, but the innovations we treat here come primarily from an injectivity criterion for conformal, local diffeomorphisms of an n-dimensional Riemannian manifold into the n-sphere. The criterion involves a generalization of the Schwarzian derivative whi...

We describe the way in which the sign of the Schwarzian derivative for a family of diffeomorphisms of the interval I affects the dynamics of an associated many-to-one skew product map of the cylinder (R/Z) × I.

The main character of the present note is the Schwarzian derivative, and we start with a brief reminder of its definition and main properties. Let f : RP → RP be a projective line diffeomorphism. For every point x ∈ RP there exists a unique projective transformation gx : RP 1 → RP whose 2-jet at x coincides with that of f . The Schwarzian derivative S(f) measures the deviation of the 3-jet jf f...

‎In the present investigation‎, ‎we give the best estimates for the norm of the pre-Schwarzian derivative $ T_{f}(z)=dfrac{f^{''}(z)}{f^{'}(z)} $ for bi-starlike functions and a new subclass of bi-univalent functions of order $ alpha $‎, ‎where‎ ‎$Vert T_{f} Vert= sup_{|z|

By considering integral bounds on the Schwarzian derivative we extend previous results on sufficient conditions for curves in euclidean spaces to be simple or unknotted. The conditions are optimal.