I have following function $$f(x,y,z)=xy\log(1+\frac{z}{xy})$$ where $x,y,z$ are all $\geq 0$. Is it a jointly concave function of $(x,y,z)$? Thank you in advance.
Edit:
After fixing $z=1$ I am trying to check whether $xy\log(1+\frac{1}{xy})$ is jointly concave with respect to $(x,y)$ or not. The determinant of the Hessian matrix for $xy\log(1+\frac{1}{xy})$ is given as follows $$1-\left(-2-xy+(1+xy)^2\log(1+\frac{1}{xy})\right)^2$$ Now if the above determinant is greater than or equal to $0$ then $xy\log(1+\frac{1}{xy})$ is a concave function.
The answer is NO. Here is a proof by contradiction.
Assume that $f$ is concave. Then, the following function must be concave on $z>0$ as a composition with a linear transformation: $$ g(z) = f(z, z, z) = z^2 \log(1+1/z). $$ From concavity, we conclude that $g''(z) \leq 0$ for all $z>0$. However, $$g''(1)=2\log(2)-5/4 \approx 0.1363 > 0,$$ which is a contradiction.