Yet another inequality: $|a+b|^p<2^p(|a|^p+|b|^p)$ Let $a$ and $b$ be real numbers and $p>0$. What is the best way to prove that $|a+b|^p<2^p(|a|^p+|b|^p)$? 

If $0<p\leq1$ you don't need any powers of $2$.  This came up in another recent question.  If $1\leq p$ you can strengthen the inequality to $|a+b|^p\leq 2^{p-1}(|a|^p+|b|^p)$ by applying convexity of the function $t\mapsto t^p$.
