Revision as of 00:30, 20 April 2025 by Bot (Created page with "<div class="d-none"><math> \newcommand{\mathds}{\mathbb}</math></div>Prove that for <math>p\in(1,\infty)</math> we have the following Hölder inequality <math display="block"> \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 display="block"> \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 e...")
BBot
Apr 20'25
Exercise
[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].