Year: 2026
Type: preprint
Abstract: In this paper, we prove that for two cyclically ordered sequences of complex numbers of unit modulus $(ξ_j)_{j=1}^{n}$ and $(ζ_j)_{j=1}^{n}$ the inequality: $$\bigg|\sum_{m=1}^{n}(ξ_{m+1}-ξ_m)ζ_m\bigg|^2+ \bigg|\sum_{m=1}^{n}(ξ_{m+1}-ξ_m)ζ_m^{-1}\bigg|^2\leqslant 4n^2\sin^2\fracπ{n}$$ 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.
Source: arXiv (Cornell University)
Authors: Božin, Vladimir, Melentijević, Petar
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212008289