Using Reverse Triangle Inequality, one can write for $x,y\in R^1$
$$ ||x|-|y||\leq |x-y| $$
Is there any suitable inequality doing the following $$ ||x|^p-|y|^p|\leq f_p(|x-y|) $$
for $1 \leq p < \infty$
Thanks for any advice.
Using Reverse Triangle Inequality, one can write for $x,y\in R^1$
$$ ||x|-|y||\leq |x-y| $$
Is there any suitable inequality doing the following $$ ||x|^p-|y|^p|\leq f_p(|x-y|) $$
for $1 \leq p < \infty$
Thanks for any advice.