Given the following:
an $(n \times z)$ matrix $A = {(a_1,a_2, \ldots ,a_n)}^{T}$ where $z \geq n$ and every $a_i$ is a $z$-dimensional row vector.
$a_i = [a_{i1}\, a_{i2}\, \ldots\, a_{iz}]$ where $ \forall j\colon a_{ij} \geq 0$.
$\forall r \in \{1,2,\ldots ,n\}\colon \sum_{i=1}^{z}a_{ri} = 1$.
$\forall p,q\colon\sum_{i=1}^{z}|a_{pi} - a_{qi}| \leq \epsilon$ where $\epsilon \ll 1$.
Find a provable upper bound on:
$$\sum_{i,j=1}^{z}\left\lvert\frac1n\cdot\sum_{k=1}^{n}[a_{ki}(a_{f(k)j} - a_{g(k)j})]\right\rvert$$
where $f$ and $g$ are permutations over the set $\{1,2,\ldots,n\}$ such that $ \forall i\colon f(i) \neq g(i)$.
I am expecting the bound to be $\epsilon^2$, but I have no idea how to prove it.