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Find the maximum value of

$$\int_{0}^{1} \int_{0}^{1} \left| f(x) - f^{-1}(y) \right|^{3} \,\mathrm d x \,\mathrm d y$$

over weakly decreasing functions $f:[0,1] \to [0,1]$, where $$f^{-1}(y) := \inf\left\{x : f(x) \le y \right\}$$

I am trying to find out how one should proceed with such a problem. I will welcome any partial solution, ideas or good tips. I even wonder if solving this problem is fairly easy or terribly complicated task.

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  • This question can be stated also as: maximize over random variables $X$ taking values in $[0,1]$ the following: $\int_{0}^{1} \mathbb{E} |X-\mathbb{P}(X\ge u)|^3 du.$ Maybe this might be of some use. –  Dec 28 '19 at 16:44
  • It is somehow similar to https://math.stackexchange.com/q/2209895/617199. –  Dec 28 '19 at 16:47
  • I would also welcome any not-trivial upper bound. –  Dec 29 '19 at 13:46

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