Let $(X, \mu)$ be a probability space, and let $p \in (1, 2)$ be arbitrary. It is known from Corollary 2.4 of this [paper](https://www.math.uwo.ca/faculty/sinnamon/pdf/holdmink.pdf) by G. Sinnamon that for any measurable $f : X \to [0, +\infty],$ we have


$$0 \leq \left( \int\_X f^p \ d\mu\right) - \left(\int\_X f \ d\mu\right)^p \leq \left\lVert f - \left(\int\_X f \ d\mu\right) \right\rVert\_p^p. \ \ \ (\*)$$ (The first inequality is an immediate consequence of Jensen's inequality.)


I would like to know how "large" this inequality can be; in other words, how far is the reverse inequality from being true?


For example, does there exist some constant $C\_p > 1$ such that for any $f$ as above and also with $\|f\|\_p < +\infty,$ we have $$\left\lVert f - \left(\int\_X f \ d\mu\right) \right\rVert\_p^p \leq C\_p \left[ \left( \int\_X f^p \ d\mu\right) - \left(\int\_X f \ d\mu\right)^p \right] \ ?$$


More generally, I am just wondering what can be said about the two sides of $(\*)$ above in relation to one another.



Mean deviation in $p$-norm for $1 < p < 2$
Let us rewrite the inequality in question as
$$E|f-1|^p\le C\_p(Ef^p-1),$$
where $E$ is the expectation with respect to $\mu$ -- assuming, without loss of generality, that $Ef=1$.


This inequality does not hold in general for any $p\in(1,2)$ and any real constant $C\_p$. Indeed, suppose e.g. that $f=1+tR$, where $P(R=1)=P(R=-1)=1/2$, $t>0$ is small, and $P:=\mu$. Then $Ef=1$,
$$E|f-1|^p=t^p,$$
and
$$Ef^p-1=\tfrac{1}{2} (1-t)^p+\tfrac{1}{2} (1+t)^p-1\sim (p-1)pt^2/2=o(t^p)=o(E|f-1|^p).$$


