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: $$\bigg|\sum_{m=1}^{n}(\xi_{m+1}-\xi_m)\zeta_m\bigg|^2+ \bigg|\sum_{m=1}^{n}(\xi_{m+1}-\xi_m)\zeta_m^{-1}\bigg|^2\leqslant 4n^2\sin^2\frac{\pi}{n}$$ holds for $n\geqslant 4$. 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.