I'm looking for two functions $f(x)$ concave and $g(y)$ convex such that the square of their ratio $h(x,y)=(f(x)/g(y))^2$ is convex. The function $f$ must map a subset of $\mathbb{R}$ to $\mathbb{R}^+$. The function $g$ must map a subset of $\mathbb{R}$ to at least the interval $[0,1]$ (could be bigger). Both functions must be strictly increasing.
I am starting to believe that there exist no such functions.