{"id":"hexagon/2610.00622","pdf":"https://hexagonmath.org/pdf/2610.00622v1","url":"https://hexagonmath.org/abs/2610.00622v1","tags":["math.CO","math.PR","msc:05E05","msc:26B25","msc:60K25","msc:90B22","McLeod inequality","closed queuing network","complete homogeneous symmetric polynomial","convexity"],"score":0,"title":"A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios","domain":"math.CA","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.CA","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro","Claude"],"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}