Let $A$ be real matrix with $M > 1$ rows and $K > 2$ columns, and each entry $a\_{m,k} \in (0,1)$, with each row summing to $1$. For all $m$
$$
\sum\_{k=1}^{K} a\_{m,k} = 1
$$
I want to find out if for any row $m$
$$
\sum\_{k=1}^{K} \frac{(a\_{m,k})^2}{\sum\_{i=1}^{M} a\_{i,k}} \geq \frac{1}{M}
$$
I believe that I have proof that it holds for $K = 2$ columns. For $K > 2$ I have tried to used gradient descent to find counterexamples but I have not found any.


* I notice $\sum\_{k=1}^{K} \sum\_{m=1}^{M} a\_{m,k} = M$.
* It seems to be a quite tight bound without further assumptions, when looking at SGD cases.
The next thing I want to look at is whether the left hand side is convex in $A$.
* Maybe I can reduce problems with $M > 2$ rows to a simpler problem with $M = 2$ rows, where some rows are summed. However, I think it is difficult to maintain the constraints on the values for the second row in that case.


I will be grateful for any pointers. Thanks.


(Context: I am thinking of the rows as agents having beliefs about a discrete rv with $K$ outcomes. Can they make a bet where they split a pot of $1$ dollar according to the fraction of probability mass they have assigned to the outcome, and if so what is the expected fraction for an agent $m$ and when if ever would it be less than $1/M$ dollar?)



Inequality for matrix with rows summing to 1
If I am not missing something, this seems a direct application of [Titu's lemma](https://brilliant.org/wiki/titus-lemma/)
$$
\sum\_{k=1}^K \frac{x\_k^2}{y\_k} \geq \frac{\left(\sum\_{k=1}^K x\_k \right)^2}{\sum\_{k=1}^K y\_k}, \quad x\_k \geq 0, y\_k > 0,
$$
which is quickly proved using Cauchy-Schwarz's inequality (as shown in the link above). Just set
$$
x\_k = a\_{m,k}, \quad y\_k = \sum\_{i=1}^M a\_{i,k}
$$
and simplify the RHS using the two relations you already wrote in your question.


