# For what value of $p$ does this hold?

For which values of $p>0$ does the inequality $|x+y|^p\le |x|^p+|y|^p$. I am thinking convexity but I am not sure. thanks

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If $x=y=1$, the inequality becomes $2^p \leq 2$, so $p \leq 1$... Do you mean $|x+y|^p \geq |x|^p+|y|^p$? –  Seirios Feb 8 '13 at 20:23
The inequality is wrong: the correct is $$|x+y|^{p} \le 2^{p-1}(|x|^{p} + |y|^{p})$$ for p>1. –  ArthurStuart Feb 8 '13 at 20:46