Define a function: $f: X \rightarrow \mathbb{R}$. The notation $\mathbb{E}_{x \sim D}[f(x)]$ denotes the mean of the function under distribution $D$, where $D$ is a continuous distribution defined over $X$.
Consider two distributions $D_1$ and $D_2$. Is there a way to provide (approximate) bounds for the following norm? $$ \|\mathbb{E}_{x \sim D_1}[f(x)] - \mathbb{E}_{x \sim D_2}[f(x)] \| $$ possibly as a function of the individual means $\mathbb{E}_{x \sim D_1}[f(x)]$ and $\mathbb{E}_{x \sim D_2}[f(x)]$, and some distance metric between the two distributions $D_1$ and $D_2$.
The best I can do is that if $f$ is non-negative and bounded, then we have $$|\sum_x \mu(x)f(x)-\sum_x \nu(x)f(x)| \le ||f||_\infty \sum_{x : \mu(x) > \nu(x)} \mu(x)-\nu(x) = ||f||_\infty tvd(\mu,\nu)$$
where I used $\mu,\nu$ instead of $D_1,D_2$, and $tvd(\mu,\nu)$ refers to their "total variation distance".