This may be a trivial question to most, but here we go:
I have two strictly concave functions, say $f(x)$ and $g(x)$.
From this can I say that a function of those two functions, $h[f(x), g(x)]$, is also strictly concave?
If this is not always true, what are the further conditions on $f(x)$ and $g(x)$ to ensure strict concavity of $h(\cdot)$?
Many thanks for the help.
Best Answer
To start with, I think you should ask yourself the same question with only one function $f(x)$. Let $f(x) = \sqrt x$ is a concave function, and $h(x) = \frac 1x$, then $h(f(x)) = \frac {1}{\sqrt x}$ is actually convexe.
So by using an example with only one function, I proved that it's not true for two functions.
Still using one function, I don't think that the conditions should only apply on $f(x)$ but also, and especially, on $h(x)$.
First you should look out for $\frac{d^2h}{dx^2}$ in terms of $h$ and $f$ and then you should be able to make a condition rule.
Only then should you try to generalise it to two functions.