Let $\mu$ be a probability measure on $\mathbb R$. Let $f$ be in $L^p(\mu)$ for $p > 1$. We know from Jensen's inequality that
$\int_\mathbb R |f|^p d\mu \geq \left(\int_\mathbb R |f| d\mu\right)^p .$
Is there any way to get an upper bound for how much these two quantities can differ? i.e. are there any known inequalities of the form
$\int_\mathbb R |f|^p d\mu - \left(\int_\mathbb R |f| d\mu\right)^p \leq (\text{something in terms of } f\text{ and }\mu)?$
Also, in the case where $p=2$, the quantity on the left above is the variance of $|f|$. Is there an interesting interpretation for other values of $p$?