This is motivated by this question on math.SE.

One way to think about Jensen's inequality is that it says that if we have some probability distribution $\mu$ (continuous or discrete) over a space $X$ and $f:X\to\mathbb{R}$ is convex, changing the distribution to a deterministic distribution $\tilde{\mu}$ concentrated at the mean of $\mu$ cannot increase the expected value of $f$, i.e. $$\mathbb{E}_{\mu}[f(x)] \geq \mathbb{E}_{\tilde{\mu}}[f(x)]$$

It seems plausible that there would be other distributions $\nu$ besides $\tilde{\mu}$ with this property (in particular, they would have to be non-deterministic). What I'm wondering is if there are nice criteria that capture what $\nu$ should look like with respect to $\mu$ and work for any convex $f$?