For instance, setting we obtain the inequality Whenever and are real (but possibly negative). More precise versions of this statement were subsequently observed by Meka-Reingold-Tal and Doron-Hatami-Hoza, who obtained estimates of the shape

On the other hand, it was observed by Gopalan and Yehudayoff that if two consecutive values are small, then this makes all subsequent values small as well. In particular, vanishing of one does not imply vanishing of all subsequent. In particular, some routine estimation then gives the order of magnitude boundįor even, thus giving a significant violation of the Maclaurin inequality even after putting absolute values around the. Here, one can verify that the elementary symmetric means vanish for odd and are equal to for even.

A key example occurs when is even, half of the are equal to, and half are equal to. Whereas Newton’s inequality works for arbitrary real, the Maclaurin inequality breaks down once one or more of the are permitted to be negative. One can think of Maclaurin’s inequality as providing a refined version of the arithmetic mean-geometric mean inequality on variables (which corresponds to the case, ). The general case of this inequality can be deduced from this special case by a number of standard manipulations (the most non-obvious of which is the operation of differentiating the real-rooted polynomial to obtain another real-rooted polynomial, thanks to Rolle’s theorem the key point is that this operation preserves all the elementary symmetric means up to ). Note that the case of this inequality is just the arithmetic mean-geometric mean inequality Valid for all and arbitrary real (in particular, here the are allowed to be negative). It can be proven as a consequence of the Newton inequality This paper concerns a variant of the Maclaurin inequality for the elementary symmetric means

I have just uploaded to the arXiv my paper “ A Maclaurin type inequality“.

Ucla sigma chi