Research Preprints · Combinatorics

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

Suvrit Sra · TU Munich, Department of Mathematics, School of CIT

Version v1 · released 5 October 2026 · 11 pages · McLeod's 1959 conjecture

PDFarXiv:2610.07302v1GitHubTimestamp

Abstract

We prove McLeod's (1959) conjecture that $(h_{\ell+r}/h_\ell)^{1/r}$ is subadditive for all $\ell\ge0$ and $r\ge1$, 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.

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Cite

@misc{sra2026mcleod-hk-v1,
  author        = {Suvrit Sra},
  title         = {{A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios}},
  year          = {2026},
  month         = oct,
  howpublished  = {\url{https://arxiv.org/abs/2610.07302v1}},
  eprint        = {2610.07302v1},
  archiveprefix = {arXiv},
  note          = {Also released as mcleod-hk v1 (5 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/mcleod-hk/v1}},
}

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