The classical Maclaurin inequality asserts that the elementary symmetric
means skβ(y)=(knβ)1β1β€i1β<β―<ikββ€nββyi1βββ¦yikββ obey the inequality sββ(y)1/ββ€skβ(y)1/k whenever 1β€kβ€ββ€n and y=(y1β,β¦,ynβ)
consists of non-negative reals. We establish a variant β£sββ(y)β£β1ββͺk1/2β1/2βmax(β£skβ(y)β£k1β,β£sk+1β(y)β£k+11β) of this inequality in
which the yiβ are permitted to be negative. In this regime the inequality is
sharp up to constants. Such an inequality was previously known without the
k1/2 factor in the denominator.Comment: 12 pages, no figures. Minor typo correction