In the $L_p$ space with the norm $\|f\|_p = \left(\int |f|^p\right)^{\frac 1p}$ one has the reverse triangle inequality:
$$| \|f\|_p - \|g\|_p | \le \| f - g\|_p $$
What I am interested is the same inequality, but all the norms are raised to the $p$ power:
$$\color{red}{| \|f\|_p^p - \|g\|_p^p | \le \| f - g\|_p^p }$$
It is in red because the inequality does not hold, as one can see by choosing $g = f/2$. My question is, is there a variation of this inequality such that it holds? Can you point me to related inequalities?