# Concavity of product and ratio of sums

Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success.

Consider the function $$f:\mathbb{R}^n\rightarrow \mathbb{R}$$ defined as $$f(x)=\left(\sum_{j=1}^{n}x_{j}\right)\left(1-\sum_{j=1}^{n}\frac{x_{j}}{1+\sum_{i=1}^{n}A_{ij}x_{i}}\right)$$ where $$0\leq x_i$$ and $$0\leq A_{ij}\leq 1$$ are constants. Assume also that the second parenthesis is always positive.

I am looking for necessary conditions on the matrix $$A$$ such that $$f$$ exhibits some form of concavity (either pseudo, log, quasi or standard). For instance, if $$n=2$$, $$A_{11}=A_{22}=0$$ and $$A_{12}=A_{21}=1$$ then $$f$$ is log concave. If all the $$A_{ij}$$ are constant then $$f$$ is also log concave. I am hoping to find that these concavity results extend to more general $$A$$'s.

• Why the vote to close? – Yemon Choi May 7 '17 at 18:53
• You might consider adding a top-level tag in order to make more people see this question. – Stefan Kohl May 7 '17 at 19:41