How to prove this inequality with exponents?

Question

Let $G = |Q_1+Q_2|^{p-1}(Q_1+Q_2) - |Q_1|^{p-1}Q_1-|Q_2|^{p-1}Q_2$ where $Q_1,Q_2:\mathbb{R}^d\to \mathbb{R}$ are positive valued functions and $p>2.$ I want to show that, $$|G| \leq K\left( |Q_1|^{p-1} |Q_2| + |Q_2|^{p-1} |Q_1|\right)$$ where $K$ is some constant that we don't really care about. I tried the following, \begin{align} |G| &\leq |Q_1+Q_2|^{p} + |Q_1|^{p}+|Q_2|^{p}\\ &\leq 2^{p-1}(|Q_1|^{p} + |Q_2|^{p}) + (|Q_1|^{p} + |Q_2|^{p})\\ &\leq (2^{p-1}+1) (|Q_1|^{p} + |Q_2|^{p}). \end{align} But this is not good. Any hints/comments would be much appreciated.

Answer 1

We have that, $$G = |Q_1+Q_2|^{p-1}(Q_1+Q_2) - |Q_1|^{p-1}Q_1-|Q_2|^{p-1}Q_2$$ and so if $|Q_2|<|Q_1|$ then if we let $x= Q_2/Q_1$ then we get that, \begin{align*} |G| &\leq |Q_1|^{p}|(1+x)^p - 1- x^p| \\ &\lesssim |Q_1|^{p} |x|\\ &\lesssim |Q_1|^{p-1}|Q_2|. \end{align*} Similarly, if $|Q_1|<|Q_2|$ then we have that, \begin{align*} |G| &\lesssim |Q_2|^{p-1}|Q_1|. \end{align*} and so we get that, $$|G|\lesssim |Q_2||Q_1|^{p-1} + |Q_1||Q_2|^{p-1}.$$