**Background**

notation: RV= random variable, $\mu=$ mean $m=$ median

Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV.

If $f(x)$ strictly convex:

$$\mu (f(x)) > f(\mu (x))\mathrm{\hspace{20mm}(1)}$$

Conversely if $-f(x)$ is strictly convex:

$$\mu (f(x)) < f(\mu (x))$$

An analogous property of the median has been presented (Merkle et al 2005, pdf).

**Motivation**

I have a nonlinear function (pdf) of positive random variables, too complex to post here, not directly pertinent to this question; I am looking for a more general answer. It is worth noting that it is, however, neither strictly concave nor convex.

In practice, I find that the function of the medians provides a much better estimate of the median of the function than does the estimate of the mean of the function from the function of the means. I am interested in learning the conditions for which this is true.

**Question**

Under what conditions will the function of a median be closer to the median of a function than the mean of a function is to a function of the mean?

Specifically for what types of $f(x)$ and $x$ is

$$|\mu (f(x)) - f(\mu (x))| > |m (f(x)) - f(m (x))|$$

*I previously asked this on here on stats.stackexchange.com, but after not receiving an answer, was advised to post here on MO.*

**References**