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

We prove McLeod’s (1959) conjecture that $(h_{\mathrm{\ell }+r}\mathrm{/}{h}_{\mathrm{\ell }}{)}^{1\mathrm{/}r}$ is subadditive for all $\mathrm{\ell }\ge 0$ and $r\ge 1$, 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 $h_k\mathrm{/}{h}_{k-1}$ 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.