Given that $x,y \in (0,1]$ and a>0, is function $f(x,y) = \frac{1}{xy}\log(xya+1)$ a convex function?
By plotting it seems to be concave. But trying to prove analytically seems a little bit tedious. I tried to estimate its Hessian matrix and find if it is positive semi-definite but it seems too complicated, and was just a fruitless try.
By observing that $\frac{1}{xy}$, is convex, is there any easier way to find that $f$ is also convex?