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PreprintCC BY 4.0arXiv

Reverse Heinz type inequality, Mather beta function and coefficient estimates

  • Vladimir Božin
  • Petar Melentijević
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Abstract

In this paper, we prove that for two cyclically ordered sequences of complex numbers of unit modulus (ξj)j=1n and (ζj)j=1n the inequality: |∑m=1n(ξm+1−ξm)ζm|2+ |∑m=1n(ξm+1−ξm)ζm−1|2leqslant 4n2sin2π/n holds for ngeqslant 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.

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Cite

BibTeXRIS
BibTeX
@misc{bozin2026reverse,
  title = {{Reverse Heinz type inequality, Mather beta function and coefficient estimates}},
  author = {Vladimir Božin and Petar Melentijević},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07864},
  eprint = {2609.07864},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07864},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Reverse Heinz type inequality, Mather beta function and coefficient estimates
AU  - Vladimir Božin
AU  - Petar Melentijević
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07864
UR  - https://arxiv.org/abs/2609.07864
AB  - 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: |∑_m=1^n(ξ_m+1−ξ_m)ζ_m|^2+ |∑_m=1^n(ξ_m+1−ξ_m)ζ_m^−1|^2leqslant 4n^2sin^2π/n holds for ngeqslant 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.
ER  -