1
$\begingroup$

I'm wondering if there is a characterization for whether $$p_\lambda(x_1, \dotsc, x_r) \geq p_\mu (x_1, \dotsc, x_r) \text{ for every $r \in \mathbb{N}$ and $x_1, \dotsc, x_r \in \mathbb{R}^+$}.$$ (Or where, instead, $x_1, \dotsc, x_r \in \mathbb{N}$.)

Here, $p_\lambda$, $p_\mu$ are power sum symmetric functions. That is, $\lambda = \lambda_1, \dotsc, \lambda_n$, $\mu = \mu_1, \dotsc, \mu_m$, $p_\lambda(x_1, \dotsc, x_r) = \prod_i\sum_j x_j^{\lambda_i}$, and $p_\mu(x_1, \dotsc, x_r) = \prod_i\sum_j x_j^{\mu_i}$.

We know that when $|\mu|=|\lambda|$, $p_\lambda(\bar x) \geq p_\mu (\bar x)$ for every $\bar x$ if and only if $\lambda \succeq \mu$ (i.e., $\lambda_1 + \dotsb +\lambda_i \geq \mu_1 + \dotsb +\mu_i$ for every $i$, assuming $\lambda_1 \geq \dotsb \geq \lambda_n$ and $\mu_1 \geq \dotsb \geq \mu_n$) , due to "Inequalities for Symmetric Means" by Cuttler, Greene, Skandera. I'm looking for a similar characterization for the case when $|\lambda|\neq|\mu|$.

Note that if $|\lambda| < |\mu|$ then $p_\lambda \not\geq p_mu$. But if $|\lambda| > |\mu|$, it could be that $p_\lambda \geq p_\mu$ (eg for $\lambda = 1\ 1$, $\mu = 2$), or $p_\lambda \not\geq p_\mu$ (eg for $\lambda = 1\ 1$, $\mu = 3$).

$\endgroup$
4
  • 3
    $\begingroup$ Huh?? Isn't the inequality obviously wrong when inhomogeneous? $\endgroup$ Jul 27, 2012 at 18:00
  • $\begingroup$ The condition $\lambda\succeq\mu$ is not invariant under permutation of indices, whereas the condition $p_\lambda(\bar x)\ge p_\mu(\bar x)$ for all $x\bar x$ obviously is, hence the two cannot be equivalent. $\endgroup$ Jul 27, 2012 at 18:52
  • $\begingroup$ Sorry, in the definition there was a missing $\lambda_1 \geq \dotsb \geq \lambda_n$. I just added it. $\endgroup$
    – D F
    Jul 27, 2012 at 19:02
  • $\begingroup$ Also: I understand form the examples that $\lambda_i$ and $\mu_i$ are positive integers, is it correct? $\endgroup$ Jul 27, 2012 at 22:40

1 Answer 1

1
$\begingroup$

In fact, if we choose $x:=(t,0,\dots,0)$, and we recall that $t^{|\lambda|}\ge t^{|\mu|}$ for all $t > 0$ only happens if $|\lambda|=|\mu|$, we conclude by the characterization you already gave, that, for any pair of multi-indices $\lambda$ and $\mu$ in $\mathbb{N}_+^{\ r}$, one has

$$p_\lambda(x)\ge p_\mu(x) \mathrm{\ \ for\ all\ } x \in\mathbb{R}_ +^n$$

if and only if

$$|\lambda|=|\mu| \mathrm{\ \ and\ } \lambda \succeq \mu\ .$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.