## Archive of posts on Nov 11, 2022

• ### Hölder means inequality

The Hölder mean of $\vec x$ with weights $\vec w$ and a parameter $p$ is defined as $M_{p,\vec w}\!\left(\vec x\right):=\left(\vec w\cdot\vec x^p\right)^{\frac 1p}$, and the value at $p=-\infty,0,+\infty$ are defined by the limits. We can prove using Jensen’s inquality that the Hölder mean increases as $p$ increases. This property can be used to prove HM-GM-AM-QM inequalities.