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I am interested in a problem and I do not know where to start looking for possible similar setting. If anyone has a direction to suggest, it would be greatly appreciated.

Consider a (finite) set $\mathcal{X}$ and a set of functions $\mathcal{F}=\{f_i : \mathcal{X} \to \mathbb{R}\}_{i=1}^k$. I am interested in conditions on $\mathcal{X}$ and $\mathcal{F}$ such that there exists an $\alpha_{\mathcal{X},\mathcal{F}}\in\mathbb{R}$ for which any random variable $X\in \mathcal{X}$ has a corresponding (non-random) element $\hat{X}\in\mathcal{X}$ satisfying $$ f_i(\hat{X}) \leq \alpha_{\mathcal{X},\mathcal{F}} \mathbb{E}\{ f_i(X)\}$$ for all $i\in[k]$.

I am certainly not hoping for a solution, but just opinions on what related subjects I could look at.

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Consider $\mathcal F = \{ f_i : \mathbb R\rightarrow \mathbb R,\; f_i \text{ convex}, \; f_i \in L^1 \}_{i=1}^k $.

Now for all $X \in \mathbb R$ random variables such that $X^- \in L^1$ define, $\hat X = E[X]$, then by Jensen's inequality you have

$$ f_i(\hat X) \leq \mathbb E[f_i(X)] \quad \alpha_{\mathbb R, \mathcal F} = 1 $$ I guess you can play around this inequality and get different version.

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