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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 $p$?

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    @user15464 Such a bound does exist, and the best I know is $\int_\mathbb{R}|f|^pd\mu-\left(\int_\mathbb{R}|f|^pd\mu\right)^p$2012-04-08

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You are asking whether a function $F$ of $f$ and $\mu$ such that $F(f,\mu)\geq \int|f|^pd\mu-\left|\int fd\mu\right|^p$ for all $f$, but $F(f,\mu):=\int|f|^pd\mu-\left|\int fd\mu\right|^p$ is allowed, and it's the best we can do.

What is interesting is too see how big can the ratio $R(f):=\frac{\int|f|^pd\mu}{\left|\int fd\mu\right|^p}$. If $\mu$ is a Borel measure absolutely continuous with respect to Lebesgue measure, take $A_n:=(-\infty,-n)\cup(n,+\infty)$. Then $\mu(A_n)$ is decreasing to $0$ and taking $f:=\chi_{A_n}$ we get $R(A_n)=\mu(A_n)^{1-p}$ which converges to $+\infty$ since $1-p<0$ and $\mu(A_n)\to 0$.

For integers values of $p$, the integer of the $p$-th power is called $p$-th moment.