Papers › A Maclaurin type inequality
A Maclaurin type inequality
Terence Tao
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The classical Maclaurin inequality asserts that the elementary symmetric means sₖ(y) = 1/nk ∑_(1 ≤i₁ < …< iₖ ≤n) y_(i₁) …y_(iₖ) obey the inequality s_ℓ(y)^(1/ℓ) ≤sₖ(y)^(1/k) whenever 1 ≤k ≤ℓ≤n and y = (y₁,…,yₙ) consists of non-negative reals. We establish a variant |s_ℓ(y)|^(1/ℓ) ≪(ℓ^(1/2))/(k^(1/2)) max(|sₖ(y)|^(1/k), |sₖ₊₁(y)|^(1/(k+1))) of this inequality in which the yᵢ are permitted to be negative. In this regime the inequality is sharp up to constants. Such an inequality was previously known without the k^(1/2) factor in the denominator.
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