If $f$ and $g$ are strictly convex and $f$ is increasing, I know that $f\circ g$ is strictly convex.
What would be an example of a function where $g\circ f$ is not strictly convex though...
I first thought of $f(x)=-x$ and $g(x)=x^2$, and then realized that $f$ and $g$ both have to be strictly convex as well. Now, I don't have any idea how to approach this.