Is $h(x,y) = g(x)f(y)$ convex, if $g(x)$ and $f(y)$ are convex in $x$ and $y$, respectively?

Question

I want to check if $h(x,y) = g(x).f(y)$ is convex, given that g(x) is convex and decreasing in $x$ on its and $f(y)$ is increasing and convex in $y$. Is there any way to check/prove the convexity of $h(x, y)$ other checking for Hessian?

Thank you.

Answer 1

It is not in general convex. Use $g(x) = -x$, $f(y) = y$, on $[0,1]$. Along the line $x=y$, the curve is $-x^2$ which is strictly concave.