$|a-b|^p \le 2^{p-1} (|a|^p + |b|^p)$ for $a,b \in \mathbb{R}, p\ge 1$

Question

$|a-b|^p \le 2^{p-1} (|a|^p + |b|^p)$ for $a,b \in \mathbb{R}, p \ge 1$.

How do we get this inequality? I can get $|a-b|^p \le 2^{p} (|a|^p + |b|^p)$ but I don't know how to get rid of a factor of $1/2$ on the RHS.

Answer 1

Using the convexity of $x\mapsto |x|^p$, you have that $$\left|\frac{x+y}{2}\right|^p\leq \frac{1}{2}|x|^p+\frac{1}{2}|y|^p,$$ for all $x,y\in \mathbb R$. Taking $x=a$ and $y=-b$ allows you to conclude.