Question

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 $\lambda = 1\ 1$, $\mu = 3$).

Answer 1

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

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