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In this post, the top answer did the following

$$ \begin{split} (1+|\xi|^2)^p &\leq (1+2|\xi-\eta|^2+2|\eta|^2)^p\\ &\leq 2^p(1+|\xi-\eta|^2+1+|\eta|^2)^p\\ &\leq c(1+|\xi-\eta|^2)^p + c(1+|\eta|^2)^p, \end{split} $$ for $p>0$, where $c=\max\{2^{p},2^{2p-1}\}$.

I am not sure how was the final inequality established. Am I missing some important identity to see this inequality?

P.S. I would usually just ask this in the comment but since that question has been answered about to a decade ago, I figured it'd be better if I ask in a separate post

junjios
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Tham
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1 Answers1

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You just have to use that $(a+b)^p \leq 2^{p-1}(a^p + b^p)$. See Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$.

almosteverywhere
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