I'm reading a paper on the classical Gagliardo-Nirenberg interpolation inequality [arXiv link](https://arxiv.org/abs/1812.04281) and there is a inequality used
$$
|v-\overline{v}|\le \left\Vert v' \right\Vert\_{r,I} \ell^{1-\frac{1}{r}}, r\ge 1
$$
where $\overline{v}:=\frac{1}{\ell}\int\_I v(x)dx$, $I$ is an interval on $R$, $v'$ is the derivative.


It looks quite simple, quite similar to the Hölder inequality, but where does the derivative come from? And I know that using the Poincaré inequality there will be a constant bound $C$, but then where does the term $\ell^{1-1/r}$ come from?


I think it should be a quite simple question but I am just stuck at it. Thanks for your comments!



A simple 1-dimensional inequality, maybe Poincaré inequality or Hölder inequality?
By the mean value theorem, $\bar v=v(t)$ for some $t\in I$. So, for all $x\in I$,
$$|v(x)-\bar v|=|v(x)-v(t)|
=\Big|\int\_t^x v'\Big|
\le\int\_I|v'|\le\|v'\|\_r\, \ell^{1-1/r};$$
the latter inequality is an instance of Hölder's inequality.


