Is the following function concave? (or log of it)

Question

I have a function

$f(x_1, x_2, ..., x_M) = \displaystyle \prod_{i = 1}^N \frac{(\sum_{j = 1}^{M} a_jx_jI_{ij})^2}{\sum_{j = 1}^{M} a_jx_j}$ in domain $\{{\bf x} \in {\bf R}^m \setminus {\bf 0} \ \mid 0 \leq x_j \leq 1\}$.

Here $a_j$ are some positive integer constants and $I_{ij}$ are indicator values (0 or 1).

I was wondering if $f$ or log $f$ is concave in this domain.

For log $f$, using trivial composition rule led me nowhere, so I was wondering if there are some other techniques I may be able to use to prove that it is indeed concave. Of course, it may not turn out to be concave after all.

Thanks!

Answer 1

It seems that the function $f(x,y)=\frac{x^2}{x+y}$ is not concave even on the set $$\{(x,y): 0<x,y<1\mbox{ and }x+y=1\}$$ and its logarithm is not concave even on the set $$\{(1/2,y): 0<y<1\}.$$