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Assume that we have a continuous random variable $X$ with bounded support $[a,b]$ such that $0

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Here are some conditions... not very interesting ones.

  1. $X$ attains only one value.
  2. $g\ge h$ everywhere.
  3. $g\le h$ everywhere.

Only the third one merits an explanation: if $g\le h$ and $E[g(X)^{-2}]\le E[h(X)^{-2}]$, then the essential range of $X$ is contained in the set $g=h$, which implies $E[g(X)]=E[h(X)]$.

If you allow $X$ to attain two values, one can construct two functions $g,h\colon [0,\infty)\to [0,\infty)$ that will break the implication. (Just make $h$ huge at one of the two values).

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    Thanks for the answer but X should be a non-degenerate continuous random variable. I added some assumptions accordingly.2012-06-07