QM-AM-GM-HM Inequalities

From the Cauchy–Schwarz inequality on real numbers, setting one vector to (1, 1, ...):

${\displaystyle \left(\sum _{i=1}^{n}x_{i}\cdot 1\right)^{2}\leq \left(\sum _{i=1}^{n}x_{i}^{2}\right)\left(\sum _{i=1}^{n}1^{2}\right)=n\,\sum _{i=1}^{n}x_{i}^{2},}$ hence ${\displaystyle \left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)^{2}\leq {\frac {\sum _{i=1}^{n}x_{i}^{2}}{n}}}$. For positive ${\displaystyle x_{i}}$ the square root of this gives the inequality.

The reciprocal of the harmonic mean is the arithmetic mean of the reciprocals ${\displaystyle 1/x_{1},\dots ,1/x_{n}}$, and it exceeds ${\displaystyle 1/{\sqrt[{n}]{x_{1}\dots x_{n}}}}$ by the AM-GM inequality. ${\displaystyle x_{i}>0}$ implies the inequality:

${\displaystyle {\frac {n}{{\frac {1}{x_{1}}}+\dots +{\frac {1}{x_{n}}}}}\leq {\sqrt[{n}]{x_{1}\dots x_{n}}}.}$