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

TL;DR

This paper extends R. Schwartz's inequality for the Schwarzian derivative, introduces discrete and planar versions, and explores geometric inequalities related to convex and star-shaped curves, with implications for billiard dynamics.

Contribution

It presents a discrete version of Schwartz's inequality, a planar Blaschke-Santalo inequality for polygons, and analyzes local minimality of central ellipses in geometric functionals.

Findings

Discrete Schwartz inequality proved.

Planar Blaschke-Santalo inequality established for polygons.

Central ellipses are local minima of certain geometric functionals.

Abstract

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.