Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves φ : D → C. The result can be considered a generalization of a classical condition for univalence of Nehari.

Period adding with the coexistence of successive at tract ing periodic orbit s is observed in the model of a chemical system when the bifurcation parameter is changed. This phenomenon is characterized by a family of one-dimensional return maps having a cusp shape with positive Schwarzian derivative that exhibits a saddle-node bifurcation.

The first group of differentiable cohomology of Diff(S 1), vanishing on the Möbius subgroup P SL(2, R) ⊂ Diff(S 1), with coefficients in modules of linear differential operators on S 1 is calculated. We introduce three non-trivial P SL(2, R)-invariant 1-cocycles on Diff(S 1) generalizing the Schwarzian derivative.

In this paper we study sharp estimates of pre-Schwarzian derivatives functions belonging to the Nehari-type classes by using techniques from differential equations. sequel, also see that a solution complex equation has special form in terms ratio hypergeometric resulting an integral representation. Finally, attempt those univalent unit disk for which image domain is unbounded John domain.

It is customary to formulate the inequalities of the "Verzerrungssatz" type for analytic functions w—f(z), schlicht in the unit circle, with reference to a specific normalization. The two normalizations mainly used are: (a) f(z) is finite in \z\ < 1 , /(O) = 0 , /'(O) = 1; (b) ƒ(z) has a pole at 2 = 0 with the residue 1. If we want to obtain inequalities which are independent of any particular ...

The families of one-dimensional maps have been the object of many studies since [Sharkovsky, 1964], for instance in [Sharkovsky et al., 1997] or in [de Melo & van Strien, 1993], whose authors also studied this subject, we have two large surveys of the enormous effort made in the last decades in this field of work. We adopt the usual definitions as for instance symbolic dynamics or topological e...

We prove that there is a residual set of families of smooth or analytic unimodal maps with quadratic critical point and negative Schwarzian derivative such that almost every non-regular parameter is Collet-Eckmann with subexponential recurrence of the critical orbit. Those conditions lead to a detailed and robust statistical description of the dynamics. This proves the Palis conjecture in this ...