Assume that we have a continuous random variable $X$ with bounded support $[a,b]$ such that $0 and two implicit functions of $X$, $g:R^+ \rightarrow R^+$ and $g:R^+ \rightarrow R^+$. We don't have the closed form solutions for the functions but only know their domain and range information. Are there any conditions over these functions or the random variable $X$ that can be specified such that $E[\frac{1}{g(X)^2}] \leq E[\frac{1}{h(X)^2}]$ would imply $E[g(X)] \geq E[h(X)]$?
Comparison of implicit functions of random variables
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probability