$f, g$ are convex and positive $\Rightarrow f(x)g(y)$ is convex?

Question

Prove or provide a counterexample: if $f$ and $g$ are real convex positive functions on some intervals, then $f(x)g(y)$ is convex.

I think if you throw in the assumption that theyr'e also increasing, the product is convex (?). This is a standard theorem in any text on convex analysis. — Stefan Smith, Oct 18 '13 at 00:00

score 3 · Accepted Answer · answered Oct 17 '13 at 21:49

3

$f(x)=x$, $g(x)=1-x$ is a counterexample on $[0,1]$.

answered Oct 17 '13 at 21:49

vadim123

score 1 · Answer 2 · answered Oct 18 '13 at 10:21

1

Since $(fg)''=f''g+2f'g'+fg''$ it's sufficient for $fg$ to be convex that additionally $f'g'\geq0$, i.e., both are decreasing or both are increasing.

answered Oct 18 '13 at 10:21

Michael Hoppe

2 Answers2