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McLeod Conjecture

The McLeod conjecture asserts that

G_(l,r)(x)=((h_(l+r)(x))/(h_l(x)))^(1/r)

is a subadditive function on vectors with nonnegative coordinates for integers l>=0 and r>=1, where h_k is the complete homogeneous symmetric polynomial of polynomial degree k and G_(l,r)(0)=0. Explicitly,

G_(l,r)(x+y)<=G_(l,r)(x)+G_(l,r)(y)

for vectors with nonnegative coordinates (McLeod 1959).

Sra (2026) reported a proof of the stronger assertion that G_(l,r) is a convex function. Since it is a homogeneous function of degree 1, it is also a subadditive function. Sra (2026) describes extensive mathematical dialogue with ChatGPT, including its suggestion of a queueing interpretation; the final tensorization argument is attributed to the author.

See also

Complete Homogeneous Symmetric Polynomial, Convex Function, Subadditive Function, Symmetric Polynomial

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References

McLeod, J. B. "On Four Inequalities in Symmetric Functions." Proc. Edinburgh Math. Soc. 11, 211-219, 1959. https://doi.org/10.1017/S0013091500021921.Sra, S. "A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios." 5 Oct 2026. https://arxiv.org/abs/2610.07302.

Cite this as:

Weisstein, Eric W. "McLeod Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/McLeodConjecture.html

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