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 $A_{ij}$ equals some constant then $f$ is also log concave. I am hoping to find that these concavity results extend to more general $A$'s.