Variations on R. Schwartz's inequality for the Schwarzian derivative

Details Variations on R. Schwartz's inequality for the Schwarzian derivative Variations on R. Schwartz's inequality for the Schwarzian derivative

2010-06-07

arxiv.org/abs/1006.1339v1

Mathematics

Differential Geometry

Scientific paper

R. Schwartz's inequality provides an upper bound for the Schwarzian derivative of a parameterization of a circle in the complex plane and on the potential of Hill's equation with coexisting periodic solutions. We prove a discrete version of this inequality and obtain a version of the planar Blaschke-Santalo inequality for not necessarily convex polygons. We consider a centro-affine analog of L\"uk\H{o}'s inequality for the average squared length of a chord subtending a fixed arc length of a curve -- the role of the squared length played by the area -- and prove that the central ellipses are local minima of the respective functionals on the space of star-shaped centrally symmetric curves. We conjecture that the central ellipses are global minima. In an appendix, we relate the Blaschke-Santalo and Mahler inequalities with the asymptotic dynamics of outer billiards at infinity.

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