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 $|a+b|^p<2^p(|a|^p+|b|^p)$? 

Well,
\begin{align*}
|a + b|^p \leq (|a| + |b|)^p &\leq 2^p \text{max}\{|a|^p, |b|^p\}\\
&\leq 2^{p - 1} (|a|^p + |b|^p - |a^p - b^p|)\\
&\leq 2^{p - 1} (|a|^p + |b|^p)
\end{align*}
