I previously asked this on here on stats.stackexchange.com, but after not receiving an answer, was advised to post here on MO.
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
$$\mu (f(x)) - f(\mu (x)) > m (f(x)) - f(m (x))$$
References