Some time ago, I ran across the following inequality (if I remember rightly): $\bigg(\sum_{i=1}^{n}v_{i}^{k_{1}}\bigg)\geq n\bigg(\frac{\sum_{i=1}^{n}v_{i}^{k_{2}}}{n}\bigg)^{\frac{k_{1}}{k_{2}}}$, for $v_{1},...,v_{n}$ positive real numbers and $k_{1},k_{2}$ real numbers, with $k_{1}\geq k_{2}$. Where did it originate? Does it have a name?

It is a direct consequence of Hölder's inequality for a discrete probability measure. In Hardy-Littlewood-Pólya it is listed as Theorem 16, and doesn't have a special name. They credit the inequality to Schlömilch (1858), Reynaud and Duhamel (1823), and Chrystal (1900).

A better way to formulate the inequality is to define the $\ell_p$ weighted mean of a finite list of positive numbers $a = (a_1, \ldots, a_n)$ by $$ \mathfrak{M}_p(a) = \left( \frac{1}{n} \sum_{i = 1}^n a_i^p \right)^{1/p}$$ and the inequality states $$ \mathfrak{M}_p \leq \mathfrak{M}_q $$ whenever $q \geq p$. It fits in the general hierarchy of inequality of means, which also includes the inequality of arithmetic and geometric means.

  • Thanks a lot for your answer! – Yakari Sep 14 at 6:34

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