Assume that $f_1$ and $f_2$ are two bounded functions from $\mathbb R^2$ to $\mathbb R$ such that $||f_1-f_2||_{L^1}\leq \varepsilon$ for some $\varepsilon>0$.
Is it possible to bound in a convenient way the following $L^1$ norm $$||f_1^n-f_2^n||_{L^1}$$ where $n\in\mathbb N$
Say $|f_1| \le M$ and $|f_2| \le M$. By the mean value theorem, $$ |f_1^n - f_2^n| \le |f_1-f_2|nM^{n-1} $$ so integrate to get $$ \|f_1^n-f_2^n\|_{L^1} \le nM^{n-1}\|f_1-f_2\|_{L^1} \le nM^{n-1}\epsilon . $$