Suppose that $(f\_m)\_m, f \subset L^p(\Omega)$ are nonnegative, such that $f\_m \rightarrow f$ in $L^p$; that is:

$$
\|f\_m - f\|\_p \rightarrow 0
$$

It is clear that $({f\_m}^r)\_m, f^r \subset L^{\frac{p}{r}}(\Omega)$, since $\int |f|^p = \int|f^r|^{\frac{p}{r}}$. Can we conclude that ${f\_m}^r \rightarrow f^r$ in $L^{\frac{p}{r}}$? That is:

$$
\left\|
{f\_m}^r - f^r
\right\|\_{\frac{p}{r}} \rightarrow 0
$$

Note: we assume $1<r<p<\infty$, and $\Omega \subset \mathbb{R}^n$ is measurable.