[math] \newcommand{\mathds}{\mathbb}[/math]

Prove that for [math]p\in(1,\infty)[/math] we have the following Hölder inequality

[[math]] \left|\frac{x_1+\ldots+x_N}{N}\right|^p\leq\frac{|x_1|^p+\ldots+|x_N|^p}{N} [[/math]]

and that for [math]p\in(0,1)[/math] we have the following reverse Hölder inequality

[[math]] \left|\frac{x_1+\ldots+x_N}{N}\right|^p\geq\frac{|x_1|^p+\ldots+|x_N|^p}{N} [[/math]]

with in both cases equality precisely when [math]|x_1|=\ldots=|x_N|[/math].