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Let $X$ be a bounded positive variable with an unknown probability density function (PDF) and $f(X)$ be a differentiable positive function. $$\begin{align*} &\min/\max &E\left[\frac{X}{f(X)}\right] =\text{ ?}\\ &\text{s.t.} &E[f(X)]=c \end{align*}$$

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I've added a bit of complicated LaTeX. It aligns the maths using the & symbols, while the ` \\ ` creates a new line. I did this because LaTeX should not be used quite as rigidly as you were trying to (putting spaces like you did depends on how and where you are viewing it, but LaTeX works best if you let the computer work out how it should be viewed. If that makes sense?). – user1729 Aug 17 at 17:26
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Does this mean that given the probability distribution of $X$ we should find the functions $f$ that achieve the min and max, or does it mean that given $f$ we should find the probability distributions that achieve the min and max, or something else? When you say "Let $f$ be this and let $X$ be that", it makes it sound as if those are fixed, so it looks like neither of the above. But if it's neither of the above, then it's not clear what the question is. – Michael Hardy Aug 17 at 17:43
Probability density function of $X$ is unknown and must be found to optimize the objective function. – Amir Kazemi Aug 17 at 18:23
@Trurl Thanks for your reply. I think this is the other form of the question you mean: math.stackexchange.com/q/457500/88652 Could you solve this by isoperimetrical methods? – Amir Kazemi Aug 20 at 12:25
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I think a start would be to re-write it to emphasize the two isoperimetric constraints: Max $\int[X/f(X)]g(X)dX$ s.t. $\int[f(X)g(X)]dX=c$ and $\int g(X)=1$, where $g(X)$ is the pdf. Then apply your favorite control theory approach, such as the Hamiltonian. – Trurl Aug 20 at 12:27
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