# Concavity of Log of Sum of Products

Given $$n, I would like to show that the function $$f_n: \mathbb{R}_+^p \rightarrow \mathbb{R}_+$$ defined by $$f_n(x_1, x_2, \ldots, x_p) = \log \left( \sum_{1 \leq i_1 < i_2 < \ldots < i_n \leq p} a_{i_1, i_2, \ldots, i_p} \prod_{k=1}^n x_{i_k} \right)$$ is concave, where $$a_{i_1, i_2, \ldots, i_p} \geq 0$$ are weighing constants. Basically the function is taking the product of any $$n$$ out of $$p$$ non-negative variables and sum (with non-negative weights $$a$$) over all $$\binom{p}{n}$$ possible choices, and then taking the log.

I run the computer simulation and see the graph looks indeed concave, but I had a hard time proving it. Thai being said, it is also possible that this function is actually not concave, in which case a counter example would suffice. Any help is greatly appreciated!