Reverse Heinz type inequality, Mather beta function and coefficient estimates

Abstract

In this paper, we prove that for two cyclically ordered sequences of complex numbers of unit modulus $({\xi }_j{)}_{j=1}^n$ and $({\zeta }_j{)}_{j=1}^n$ the inequality: $$|\sum _{m=1}^n({\xi }_{m+1}-{\xi }_m){\zeta }_m{|}^2+|\sum _{m=1}^n({\xi }_{m+1}-{\xi }_m){\zeta }_m^{-1}{|}^2⩽4n^2{\sin }^2\frac{\pi }{n}$$ holds for $n⩾4$. As a consequence, for $n=4,$ we disprove Hall's conjecture on Heinz type inequality for harmonic self-mappings of the unit disk. The connection with the theory of elliptic billiards via the Mather beta function will also be given.

11 pages, preliminary version; we intend to include new results in the next version