According to the wikipedia page, the function $\varphi$ must be convex.
I would like to define a function $\varphi(x)$ where $$\varphi(f(x)) = f(x)*h(x)$$
This is because the integral at hand is $\int_0^1 f(x)h(x) \,dx$ and I want to use Jensen's inequality to pull the $h(x)$ outside of the integral, while leaving the $f(x)$ inside.
I can show that $\varphi(x) = x*h(x)$ is convex over $x$. However, I know that's not exactly what I'm doing in the above equation. The first equation is more like $\varphi(t) = t*h(x)$, which I don't think I can say if it's in general convex, since there are two variables (and do I need to show it's convex over $t$? or over $x$?). However, the function is convex over $x$ when I put in the value needed for the integral at hand, which is $f(x)$.
I do not want to use $\varphi(x) = f(x)*h(x)$ because that is not convex, and because I want to keep the $f(x)$ on the inside of the integral.
Can anyone help me understand this issue of convexity of $\phi$ better, and if the function is ok the way I defined it?
Thank you so much!