A variant of Maclaurin’s inequality

The classical Maclaurin inequality asserts that the elementary symmetric means





s
k

(
y
)
≔

1



(



n
k



)






∑



1

≤



i
1

>

⋯


>

i
k


≤


n



y


i
1




…



y


i
k




\begin{equation*} s_k(y) ≔\frac {1}{\binom {n}{k}} \sum _{1 \leq i_1 > \dots > i_k \leq n} y_{i_1} \dots y_{i_k} \end{equation*}



obey the inequality





s

ℓ



(
y

)

1

/


ℓ





≤



s
k

(
y

)

1

/

k



s_\ell (y)^{1/\ell } \leq s_k(y)^{1/k}



whenever




1

≤


k

≤



ℓ



≤


n

1 \leq k \leq \ell \leq n



and




y
=
(

y
1

,

…


,

y
n

)

y = (y_1,\dots ,y_n)



consists of nonnegative reals. We establish a variant





|


s

ℓ



(
y
)


|



1

ℓ






≪





ℓ



1

/

2



k

1

/

2



max
(

|


s
k

(
y
)


|



1
k



,

|


s

k
+
1


(
y
)


|



1

k
+
1




)

\begin{equation*} |s_\ell (y)|^{\frac {1}{\ell }} \ll \frac {\ell ^{1/2}}{k^{1/2}} \max (|s_k(y)|^{\frac {1}{k}}, |s_{k+1}(y)|^{\frac {1}{k+1}}) \end{equation*}



of this inequality in which the




y
i

y_i



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


k^{1/2}



factor in the denominator.