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PreprintCC BY 4.0arXiv

A new arrangement inequality (revised and extended version)

  • Mohammad Javaheri
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Abstract

We study the cyclic inequality ∑i=1n xi ∑i=1n xia xi+1b ≤ (∑i=1n xi(1+a+b)/2)2, where x1,…,xn≥ 0 and indices are taken modulo n. Let N(a,b) be the largest n for which the inequality holds for all nonnegative xi. We show that N(a,b) is bounded above by an explicit quantity N*(a,b) obtained from the second variation at the constant vector, and we conjecture that N(a,b)=N*(a,b). For a=b=1 we prove that the inequality holds if and only if n≤ 8, which settles the case left open in the original version of this paper. For n=4 the inequality holds whenever 1,a,b are the sides of a triangle, with an explicit remainder term. We show that N is monotone along rays emanating from the line a+b=1; this corrects and extends an earlier monotonicity statement. Finally, any n nonnegative numbers can be arranged so that the inequality holds; for n≠ 6 this is achieved by one of n−1 prescribed arrangements, which yields a volume bound for every such n.

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Cite

BibTeXRIS
BibTeX
@misc{javaheri2026arrangement,
  title = {{A new arrangement inequality (revised and extended version)}},
  author = {Mohammad Javaheri},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33653},
  eprint = {2610.33653},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33653},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A new arrangement inequality (revised and extended version)
AU  - Mohammad Javaheri
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33653
UR  - https://arxiv.org/abs/2610.33653
AB  - We study the cyclic inequality ∑_i=1^n x_i ∑_i=1^n x_i^a x_i+1^b ≤ (∑_i=1^n x_i^(1+a+b)/2)^2, where x_1,…,x_n≥ 0 and indices are taken modulo n. Let N(a,b) be the largest n for which the inequality holds for all nonnegative x_i. We show that N(a,b) is bounded above by an explicit quantity N^*(a,b) obtained from the second variation at the constant vector, and we conjecture that N(a,b)=N^*(a,b). For a=b=1 we prove that the inequality holds if and only if n≤ 8, which settles the case left open in the original version of this paper. For n=4 the inequality holds whenever 1,a,b are the sides of a triangle, with an explicit remainder term. We show that N is monotone along rays emanating from the line a+b=1; this corrects and extends an earlier monotonicity statement. Finally, any n nonnegative numbers can be arranged so that the inequality holds; for n≠ 6 this is achieved by one of n−1 prescribed arrangements, which yields a volume bound for every such n.
ER  -