Convergence in $L^p$ of $f_n$ implies convergence in $L^1$ of $|f_n|^p$ and $f_n^p$

If $a,b\in \mathbb{R}$ and $p\ge1$, then it follows from the mean value theorem that $$ |a^p-b^p|\le p\max(|a|,|b|)^{p-1}|a-b|\le p(|a|+|b|)^{p-1}|a-b|. $$ Let $q$ be the conjugate exponent of $p$. Then $$ \int\bigl|\,|f_n|^p-|f|^p\,\bigr|\le p\int\bigl(|f_n|+|f|\bigr)^{p-1}|f_n-f|\le p\Bigl(\int\bigl(|f_n|+|f|\bigr)^{(p-1)q}\Bigr)^{1/q}\Bigl(\int|f_n-f|^{p}\Bigr)^{1/p}, $$ that is $$ \||f_n|^p-|f|^p\|_1\le p(\||f_n|+|f|\|_p)^{p/q}\|f_n-f\|_p\le p(\|f_n\|_p+\|f\|_p)^{p/q}\|f_n-f\|_p, $$ proving that $|f_n|^p\to|f|^p$ in $L^1$.

Observe that we have proved that the operators $T_1, T_2\colon L^p\to L^1$ defined by $T_1(f)=|f|^p$ and $T_2(f)=f^p$( when $T_2$ makes sense) are locally Lipschitz.

If we assume that the result is not true, we would be able to find a subsequence $\{f_{n_k}\}$ and a $\delta>0$ such that $\lVert |f_{n_k}|^p-|f|^p\rVert_{L^1}\geq \delta_0$. Taking if necessary a further subsequence, we can assume that $f_{n_k}$ converges to $f$ almost everywhere. We denote $g_k:=|f_{n_k}|^p$, and let $h_k:=|g_k|+|f|^p-|g_k-|f|^p|$. Then almost everywhere, $h_k$ converges to $2|f|^p$, and since $h_k\geq 0$, by Fatou lemma: $$2\lVert f\rVert_{L^p}^p\leq \liminf_{k\to +\infty}\int (|g_k|-|g_k-|f|^p|)+\int|f|^p$$ and since $\int g_k\to \int |f|^p$ we have that $\limsup_{k\to \infty}\int |g_k-|f||^p=\limsup_{k\to \infty}\int ||f_{n_{k}}|^p-|f|^p|=0$. I followed a method similar to here.

For $f_n^p$ just choose $g_k =f_{n_k}^p$.