Reverse Heinz type inequality, Mather beta function and coefficient estimates

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: $$\mid \sum _{m=1}^n({\xi }_{m+1}-{\xi }_m){\zeta }_m{\mid }^2+\mid \sum _{m=1}^n({\xi }_{m+1}-{\xi }_m){\zeta }_m^{-1}{\mid }^2⩽4n^2{\sin }^2\frac{\pi }{n}$$ 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.