Product of concave functions is concave

Question

I have something like this $$ h(x) = f(x) g(x) $$ where $f$ and $g$ are some nice functions such that $f$ is $[0,1]$-bounded, concave function and $g$ is some negative, decreasing, concave function.

Is $h$ concave? If not, then can we impose more conditions on $f$ and $g$ such that $h$ is concave?

Answer 1

No. Let $f$ be decreasing concave and bounded and $g(x)=-x$. Then $fg$ is actually convex for $x >0$. You can see this by computing the second derivative.

Answer 2

Let's take $$ f(x)=g(x)=x, $$ which is concave (but not strictly concave).

Then $h(x)=x^2$, which is convex.

NB I take the definition of concavity from Wikipedia. It might be the other way round in your textbook.

You can then easily find strictly concave examples. Take $$ f(x)=g(x)=x^{2/3}, $$ which are both concave, but $f\times g$ is clearly convex.