Here is an old result of Siegel that is related to your question.
Set
$$ s=s(a_1,\dotsc, a_n)=\frac{1}{n} (a_1+\cdots +a_n), $$
$$ p= p(a_1,\dotsc, a_n), $$
$$ \Delta= \Delta(a_1,\dotsc, a_n)=\prod_{i,j}(a_i-a_j)^2. $$
The AM-GM inequality reads
$$\frac{s^n}{p}\geq 1. $$
Observe that $s$ is homogeneous of degree $1$, $p$ is homogeneous of degree $n$ and $\Delta$ is homogeneous of degree $n(n-1)$ in the variables $a_j$. In particular, the ration
$$ R= \frac{p^{n-1}}{\Delta} $$
is homogeneous of degree $0$. Note that $\Delta=0$ when two of the numbers $a_j$ are equal. In particular, large $\Delta $ would mean that the numbers are "far from being equal". Equivalently the larger $\Delta$ is, the more "dispersed" are the numbers $a_j$.
One can ask how dispersed can the numbers $a_j$ be given that $s$ and $p$ are fixed.
In other words we ask to find
$$\max \Delta(a_1,\dotsc, a_n)$$
given that
$$s(a_1,\dotsc, a_n)=s_0,\;\;p(a_1,\dotsc, a_n)=p_0. $$
This constrained maximum exists and can be described explicitly as the discriminant of a certain Laguerre polynomial. I will denote it by $\Delta_\max(s_0,p_0)$.
I will set
$$ \rho=\rho(s_0,p_0)= \frac{p_0^{n-1}}{\Delta_\max(s_0,p_0)}. $$
Then there exists an explicit but very complicated strictly decreasing continuous function
$$ F_n: (0,\infty)\to (1,\infty) $$
such that
$$\lim_{t\to\infty} F_n(t)=1, $$
$$ \frac{s_0^n}{p_0}= F_n(\rho)= F_n\left( \frac{p_0^{n-1}}{\Delta_\max(s_0,p_0)}\right). $$
A few things a bout the function $F_n$. It is described as a composition $Q_n\circ P_n^{-1}$, were
$$ Q_n: (0,\infty)\to (1,\infty) $$
is a strictly decreasing very explicit rational function and
$$P_n:(0,\infty)\to (0,\infty) $$
is a very explict and strictly increasing polynomial such that $P_n(0)=0$. This implies the sharper inequality
$$ s(a_1, \dotsc, a_n)^n \geq F\left(\frac{p(a_1,\dotsc, a_n)^{n-1}}{\Delta(a_1,\dotsc, a_n)}\right)p(a_1,\dotsc, a_n), $$
with equality iff
$$ \Delta(a_1,\dotsc,a_n)=\Delta_\max(s,p). $$
For more details see Sec. 8.6 of the beautiful book Special Functions by G.E. Andrews, R. Askey, R. Roy.
