Suppose I have an $n\times m$ matrix of nonnegative real numbers
$$ \begin{bmatrix} x_{1,1} & x_{1,2} & \cdots & x_{1,n} \\ x_{2,1} & x_{2,2} & \cdots & x_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ x_{m,1} & x_{m,2} & \cdots & x_{m,n} \\ \end{bmatrix} $$
where the column totals are denoted $x_{\cdot j}=\sum_i x_{ij}$ and the row totals are denoted $x_{i\cdot}=\sum_j x_{ij}$.
I would like to approximate the Herfindahl---Hirschman index of the column totals,
$$ \frac{\sum_j x_{\cdot j}^2}{(\sum_j x_{\cdot j})^2}, \tag{1}\label{eq1} $$
by a weighted average of row-specific Herfindahl---Hirschman indices:
$$ \sum_i \left(\frac{x_{i\cdot}}{\sum_i x_{i\cdot}}\right) \left(\frac{\sum_j x_{ij}^2}{(\sum_j x_{ij})^2}\right) \tag{2}\label{eq2} $$
This is a first-order (linear) approximation. My question is: can we say anything about the approximation error, like whether it is negative (indicating concavity) and/or bounded, in general or perhaps as $n\to\infty$?
The Herfindahl---Hirschman index can also be viewed as a sum of squared probabilities, or the collision probability.
What I have tried so far
For the special case of $m=2,n=3$, I worked out the approximation error (i.e. \eqref{eq1} minus \eqref{eq2}) to be
$$ -2\left(\sum_i\sum_j x_{ij}\right)^{-2} \left[ x_{11}x_{22}+x_{11}x_{23}+x_{12}x_{21}+x_{12}x_{23}+x_{13}x_{21}+x_{13}x_{22} \\ -(x_{11}x_{12}+x_{11}x_{13}+x_{12}x_{13})\frac{x_{21}+x_{22}+x_{23}}{x_{11}+x_{12}+x_{13}} \\ -(x_{21}x_{22}+x_{21}x_{23}+x_{22}x_{23})\frac{x_{11}+x_{12}+x_{13}}{x_{21}+x_{22}+x_{23}} \right], $$
which shows that we have a negative term on the outside of the square brackets, and a term of unknown sign and magnitude on the inside of the square brackets.
I have a specific application in mind, with $m=6$ and $n=162$. When I try the approximation on many such arrays in my data, I end up with the performance in the figure below. The approximation errors are all negative, suggesting that the Herfindahl---Hirschman index is concave.
This question is similar to, but more specific than, this unanswered one: Decomposing Simpson's Diversity Index/Herfindahl Index.