In this note we present an application of the Schwarzian derivative. By exploiting some properties of the Schwarzian derivative, we solve the equation appearing in the gravity-dilaton-antisymmetric tensor system. We also mention that this method can also be used to solve some other equations. The Schwarzian derivative appears naturally in the transformation law of the two-dimensional stress-ene...

The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to conformal mappings of polygons bounded by circular arcs. More recently, Nehari [5, 6, 7] and others have developed important criteria for global univalence in terms of the Schwarzian derivative, exploiting...

Because of all the known integrable models possess Schwarzian forms with Möbious transformation invariance, it may be one of the best way to find new integrable models starting from some suitable Möbious transformation invariant equations. In this paper, the truncated Painlevé analysis is used to find high dimensional Schwarzian derivatives. Especially, a three dimensional Schwarzian derivative...

We consider two theorems formulated in the derivation of the Quantum Hamilton–Jacobi Equation from the EP. The first one concerns the proof that the cocycle condition uniquely defines the Schwarzian derivative. This is equivalent to show that the infinitesimal variation of the stress tensor “exponentiates” to the Schwarzian derivative. The cocycle condition naturally defines the higher dimensio...

In a recent paper, we verifed a conjecture of Mej́ıa and Pommerenke that the extremal value for the Schwarzian derivative of a hyperbolically convex function is realized by a symmetric hyperbolic “strip” mapping. There were three major steps in the verification: first, a variational argument was given to reduce the problem to hyperbolic polygons bounded by at most two hyperbolic geodesics; secon...

H. Sato introduced a Schwarzian derivative of a contactomorphism of R 3 and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a contact projective structure is defined to be a cocycle of the contactomorphism group taking values in the space...

Let the functions q1 be analytic and let q2 be analytic univalent in the unit disk. Using the methods of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic function f are obtained so that either q1 z ≺ zf ′ z /f z ≺ q2 z or q1 z ≺ 1 zf ′′ z /f ′ z ≺ q2 z . As applications, sufficient conditions are determined relati...

In the study of one dimensional dynamical systems one often assumes that the functions involved have a negative Schwarzian derivative. In this paper we consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functi...

H. Sato introduced a Schwarzian derivative of a contactomorphism of R 3 and and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a contact projective structure is defined to be a cocycle of the contactomorphism group taking values in the s...

The aim of this note is to relate the classical Schwarzian derivative and the geometry of Lorentz surfaces of constant curvature. 1. The starting point of our investigations lies in the following remark (joint work with L. Guieu). Consider a curve y = f(x) in the Lorentz plane with metric g = dxdy. If f (x) > 0, then its Lorentz curvature can be computed : ̺(x) = f (x) (f (x)) and enjoys the qui...