Maclaurin's Inequality¶
The descending chain of root-normalized elementary symmetric means of a finite nonnegative real vector.
Core Idea¶
For nonnegative numbers \(a_1,\ldots,a_n\), average all products of \(k\) distinct entries to get the \(k\)-th elementary symmetric mean \(s_k=e_k/\binom nk\). Maclaurin's inequality says \(s_1\geq s_2^{1/2}\geq\cdots\geq s_n^{1/n}\). The endpoints are the arithmetic and geometric means, but the intermediate comparisons are the distinctive content.[^ref-6121f19dd244]
Scope of Application¶
The theorem bounds higher-degree symmetric means from earlier ones. For a polynomial whose roots are all nonnegative real numbers, those means are its normalized alternating coefficients; the same chain therefore bounds its coefficient profile. A polynomial with signed roots does not inherit the classical chain merely because its roots are real.[^ref-6121f19dd244]
Clarity¶
Both normalizations matter: dividing by \(\binom nk\) removes the number of \(k\)-subsets, and taking the \(k\)-th root makes different-degree products comparable. Newton's inequalities provide an adjacent log-concavity route to the result, but are not the same chain.[^ref-6121f19dd244]
Manages Complexity¶
Instead of bounding every subset product separately, one nonincreasing sequence controls all degrees. A known upper bound for one root mean bounds every later root mean—provided all inputs are nonnegative. Signed entries can cancel and invalidate that inference.[^ref-6121f19dd244]
Abstract Reasoning¶
For \((1,2,3)\), \(e_1=6\), \(e_2=11\), \(e_3=6\), hence the compared quantities are \(2\), \(\sqrt{11/3}\), and \(\sqrt[3]6\), in decreasing order. Equal inputs make the whole chain equal. Yet \((1,0,0)\) has \(s_2=s_3=0\): equality at one zero tail link alone does not prove all inputs equal.[^ref-6121f19dd244]
Knowledge Transfer¶
The vector and nonnegative-root polynomial versions literally share the same \(s_k\). Combinatorial extensions use related reasoning but require their own hypotheses, and broad talk of “higher-order combinations” outside this algebra is analogy.
[^ref-6121f19dd244]: Terence Tao, “A Maclaurin type inequality” (2023), abstract and §1, especially equations (1.1)–(1.4). [^ref-78003541f474]: Vladimir Nikiforov, “An extension of Maclaurin's inequalities” (2006; v3 2013), abstract.
Neighborhood in Abstraction Space¶
Maclaurin's Inequality sits in a sparse region of the domain-specific corpus (78th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Polynomials & Algebraic Invariants (20 abstractions)
Nearest neighbors
- AM–GM Inequality — 0.86
- Gram Matrix — 0.85
- Semiperfect Number — 0.82
- Daniell Integral — 0.81
- Maharam Algebra — 0.81
Computed from structural-signature embeddings · 2026-10-08