Given a product of $K$ Geometric distributions and a sum over these products, that is: $$ f(\boldsymbol{x}=(x_1,...x_k); v) = \sum_{c_1 + ...+ c_K = v} \prod_k (1- x_k) x_{k}^{ c_{k}} $$
where $c_1 + ...+ c_K = v$ denotes all the integer partitions of $v$. That is, all combinations of $k$ integers that sum up to $v$. $x_k \in [0,1]$
It seems, empirically, that $f(x)$ is, at least, unimodal. The figure shows the case for $v=10$ and $K=2$.
I would like to demonstrate analytically that the function is log-concave or, at least, unimodal (or with K! symmetric modes that reflect the symmetries on the integer partitions).
To check log-concavity I'm trying to compute the Hessian matrix of $\ln f(x)$, but it quickly becomes very hard to deal with (and especially seeing whether it is always negative) I wonder whether I'm doing it the right way or there is some smarter way to check unimodality.
I have read, for instance, that Geometric distributions are log-concave, and therefore the product of $K$ Geometrics is log-concave. Its convolution would be also log-concave, but I think my sum is not a convolution. Yet, I have the feeling that this sum over the partitions might respect the unimodality.
Any hints?
Note: this question, though much more general, might be related. The geometric distribution is known to be log-concave. Thus the product over $K$ is log-concave. The mistery is why this sum (or what kind of sums) preserves log-concavity.