You want Maclaurin's inequality.  Given $n$ positive numbers $a_1, a_2,\dots,a_n$, 
write 
$$
(x+a_1)(x+a_2)\cdots(x+a_n) = x^n + S_1x^{n-1} + \cdots + S_{n-1}x + S_n, 
$$
so $S_i$ is the $i$-th elementary symmetric function of $a_1,\dots,a_n$. 
For $i = 1,\dots,n$, set $A_i = S_i/\binom{n}{i}$.  When 
$n = 2$, $A_1 = (a_1+a_2)/2$ and $A_2 = a_1a_2$.  Maclaurin's inequality is that
$$
A_1 \geq \sqrt{A_2} \geq \sqrt[3]{A_3} \geq \cdots \geq \sqrt[n]{A_n},
$$
where the inequality signs are all strict unless $a_1,\dots,a_n$ are all equal. 
When $n = 2$ this is the arithmetic-geometric mean inequality.

From a list of $n$ positive numbers $a_1,\dots,a_n$ we have produced a list of $n$ new positive numbers $A_1,\sqrt{A_2},\dots,\sqrt[n]{A_n}$.  Repeat the construction. 
When $n = 2$ this is the arithmetic-geometric mean recursion. 

Theorem: All the terms in the list tend to the same limit.

This was studied by Meissel in 1875 for $n = 3$.