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arXiv · 2501.07613

A general form of Newton-Maclaurin type inequalities

Changyu Ren
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Abstract

In this paper, we extend the classical Newton-Maclaurin inequalities to functions $S_{k;s}(x)=E_k(x)+\dsum_{i=1}^s \al_i E_{k-i}(x)$, which are formed by linear combinations of multiple basic symmetric mean. We proved that when the coefficients $\al_1,\al_2,\cdots,\al_s$ satisfy the condition that the polynomial $$t^s+\al_1 t^{s-1}+\al_2 t^{s-2}+\cdots+\al_s $$ has only real roots, the Newton-Maclaurin type inequalities hold for $S_{k;s}(x)$.

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Changyu Ren. 2025-01-13. A general form of Newton-Maclaurin type inequalities. https://arxiv.org/abs/2501.07613

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