Suppose two convex functions $f(x)$ and $g(x)$ that are everywhere strictly positive and are such that $g^{-1}(f(x))$ is also convex ($g$ is convex with respect to $f$). Let $x$ be a random variable in $L^1$. Jensen's inequality can be used to show:
$$\frac {E[f(x)]}{f(E[x])} \ge 1$$
I would like to show:
$$\frac {E[g(x)]}{g(E[x])} \ge \frac {E[f(x)]}{f(E[x])}$$
Is this true? If not, what about showing:
$$\frac {E[f(x^n)]}{f(E[x^n])} \ge \frac {E[f(x)]}{f(E[x])}$$
where $n > 1$?