Inequalities

QM-AM-GM-HM Inequalities

Suppose that $x_1,x_2,\dots ,x_n$ are positive real numbers, then $\begin{array}{r}0<\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}}\le \sqrt[n]{x_1{x}_2\cdots x_n}\le \frac{x_1+x_2+\cdots +x_n}{n}\le \sqrt{\frac{x_1^2+x_2^2+\cdots +x_n^2}{n}}.\end{array}$