## Inequalities for pth powers, where 0<p<infinity

There are some useful inequalities for $|x+y|^p$, where p is a number ranging from 0 to infinity. These are the top 3 useful inequalities (note some of them only work for p less than 1, or p greater than 1).

1)
For $a,b\in\mathbb{R}$, $|a+b|^p\leq 2^p(|a|^p+|b|^p)$, where $0.

Proof:
\begin{aligned} |a+b|^p&\leq(|a|+|b|)^p\\ &\leq(2\max\{|a|,|b|\})^p\\ &=2^p(\max\{|a|,|b|\})^p\\ &\leq 2^p(|a|^p+|b|^p). \end{aligned}

2)
If $0, $|a+b|^p\leq|a|^p+|b|^p$ for all $a,b\in\mathbb{R}$.

Proof:
$\displaystyle 1=\frac{|a|}{|a|+|b|}+\frac{|b|}{|a|+|b|}\leq\left(\frac{|a|}{|a|+|b|}\right)^p+\left(\frac{|b|}{|a|+|b|}\right)^p=\frac{|a|^p+|b|^p}{(|a|+|b|)^p}.$
Hence $|a+b|^p\leq(|a|+|b|)^p\leq|a|^p+|b|^p$.

3)
For $a,b\in\mathbb{R}$, $|a+b|^p\leq 2^{p-1}(|a|^p+|b|^p)$ for $1\leq p<\infty$.

Proof:
By convexity of $|x|^p$ for $1\leq p<\infty$, $\displaystyle \left|\frac 12 a+\frac 12 b\right|^p\leq\frac 12 |a|^p+\frac 12 |b|^p.$
Multiplying throughout by $2^p$ gives $\displaystyle |a+b|^p\leq 2^{p-1}(|a|^p+|b|^p).$