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A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios

  • Suvrit Sra
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Abstract

We prove McLeod's (1959) conjecture that (hℓ+r/hℓ)1/r is subadditive for all ℓ≥0 and r≥1, where hk is the complete homogeneous symmetric polynomial of degree k (J. B. McLeod, "On four inequalities in symmetric functions", Proc. Edinburgh Math. Soc., 1959). We first prove that the ratio hk/hk−1 is convex, and our proof takes an unexpected detour through queuing theory. The geometric mean of r consecutive such ratios equals McLeod's conjectured function, but geometric means of convex functions need not be convex. Our second and decisive step is a tensorization trick that couples neighboring ratios tightly enough to make their geometric mean convex.

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Cite

BibTeXRIS
BibTeX
@misc{sra2026proof,
  title = {{A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios}},
  author = {Suvrit Sra},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07302},
  eprint = {2610.07302},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07302},
}
RIS
TY  - UNPB
TI  - A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios
AU  - Suvrit Sra
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07302
UR  - https://arxiv.org/abs/2610.07302
AB  - We prove McLeod's (1959) conjecture that (h_ℓ+r/h_ℓ)^1/r is subadditive for all ℓ≥0 and r≥1, where h_k is the complete homogeneous symmetric polynomial of degree k (J. B. McLeod, "On four inequalities in symmetric functions", Proc. Edinburgh Math. Soc., 1959). We first prove that the ratio h_k/h_k−1 is convex, and our proof takes an unexpected detour through queuing theory. The geometric mean of r consecutive such ratios equals McLeod's conjectured function, but geometric means of convex functions need not be convex. Our second and decisive step is a tensorization trick that couples neighboring ratios tightly enough to make their geometric mean convex.
ER  -