Given a tuple of positive numbers tuple $X = (x_1, x_2, \dots,)$, define the $k$th elementary symmetric mean of the first $n$ entries to be $$ S(X, n, k) := \frac{\displaystyle\sum_{i_1 < \cdots < i_k \leq n} x_{i_1}x_{i_2}\cdots x_{i_k}}{\dbinom{n}{k}} $$

Note that $S(X,n,1)$ is the arithmetic mean and $S(X,n,n)^{1/n}$ is the geometric mean of the first $n$ entries of $X$.

I'm interested in the behavior of these means when $X$ is a periodic sequence. The AM and GM are independent of $n$, as long as $n$ is a multiple of the period. While this does not hold for the other symmetric means, I still believe we have monotonicty. Specifically, I think that when $X=(x_1, \dots, x_L, x_1, \dots)$ is periodic with period $L$, we have the following inequality for each $k \in \mathbb{N}$ and $c \in \{1/L, 2/L, \dots, 1\}$

$$ S(X, kL, ckL)^{1/ckL} \geq S(X, (k+1)L, c(k+1)L)^{1/c(k+1)L} $$

I have verified this for numerous specific cases in Mathematica, but I don't know how to prove this in general. If it helps at all, I'm mostly interested in the case when the $x_i$ are positive integers. Any ideas?