inequalities – Mathtuition88

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 $0<p<\infty$ .

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 $0<p<1$ , $|a+b|^p\leq|a|^p+|b|^p$ 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 $\displaystyle |a+b|^p\leq 2^{p-1}(|a|^p+|b|^p).$