# Sufficient or necessary conditions for dominance of expectation of min(c, random-variable)

If $D$ and $D'$ are non-negative random variables, what are sufficient and/or necessary conditions for $E[D] \geq E[D']$ to imply $E[\min(c,D)]\geq E[\min(c,D')]$ for any $c\geq 0$.

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Although it is difficult to discern what is being asked here, one might be concerned with the identities $$D=\int_0^{+\infty}[D\geqslant x]\cdot\mathrm dx \quad\text{and}\quad \min(c,D)=\int_0^{c}[D\geqslant x]\cdot\mathrm dx,$$ which yield $$\mathrm E(D)=\int_0^{+\infty}\mathrm P(D\geqslant x)\cdot\mathrm dx \quad\text{and}\quad \mathrm E(\min(c,D))=\int_0^{c}\mathrm P(D\geqslant x)\cdot\mathrm dx.$$