conditions for Product of convex and concave function to be concave

Question

This has been asked (in a a way) here, but I don't understand how the accepted answer addresses this. It links a theorem and says that the theorem gives conditions, but I don't see how, so perhaps I am simply not understanding how to apply the theorem.

Anyway, to restate the question, suppose I have two functions, $f(x),g(x)$, and $f(x)$ is convex and $g(x)$ is concave. What are some conditions that will guarantee that $f(x)g(x)$ is concave?

---

My thoughts: If I want $f(x)g(x)$ to be concave, then I need $-f(x) g(x)$ to be convex.

But $-f(x)g(x) = f(x)\times (-g(x))$, and $-g(x)$ is convex, so now I have the product of two convex functions. Then do I just apply the results of the theorem here to this?

Is it possible to relax the condition that both convex functions be positive?

For example, $Log(x+1)*\frac{5-x}{5}$ is concave from 0 to 5, and $Log(x+1)$ is concave and $\frac{5-x}{5}$ is convex (granted it is also concave)$

Perhaps this suggests that Log concavity might be of use? Maybe if $f$ is Log-concave and $g$ is log-convex their product will be concave?

Answer 1

Basically, the point is this. Let's look at the smooth case.

$$(fg)'' = f'' g + 2 f' g' + f g''$$ So you have three terms on the right, which are each positive or negative depending on $f$ and $g$ being positive or negative, increasing or decreasing, convex or concave.

If you want a nice condition that makes $fg$ concave, you'll want its second derivative $\le 0$, and it would be nice if each term on the right was $\le 0$: otherwise you'd have to compare the sizes of the positive and negative terms, and things get messy. So if $f$ is convex you'll want $g \le 0$ to make the first term $\le 0$; you'll want one of $f$ and $g$ nonincreasing and the other nondecreasing to make the second $\le 0$; and if $g$ is concave you'll want $f \ge 0$ to make the third term $\le 0$.