Consider the function $$ f(x)=\left(\sum_{j=1}^{n}x_{j}\right)\left(1-\sum_{j=1}^{n}\frac{x_{j}}{1+\sum_{i=1}^{n}A_{ij}x_{i}}\right) $$ where $0\leq x_i\leq 1$ and $0\leq A_{ij}\leq 1$. Assume also the the second parenthesis is always positive.
I'm interested in the concavity of this function. In particular, what additional conditions (if any) must be imposed on $A$ to insure the log concavity of $f$? Of course, $A_{ij}=0$ works. It seems that $f$ is also log concave if $A_{ij}=c$ where $c$ is some constant. Any help would be appreciated!