$$\begin{aligned}H{n}&=\frac{n}{\sum\limits{i=1}^{n}\frac{1}{x{i}}} \\ &= \frac{n}{\frac{1}{x{1}}+ \frac{1}{x{2}}+ \cdots + \frac{1}{x{n}}}G{n} \\ &=\sqrt[n]{\prod \limits{i=1}^{n}x{i}}= \sqrt[n]{x{1}x{2}\cdots x{n}}A{n} \\ &=\frac{1}{n}\sum \limits{i=1}^{n}x{i}=\frac{x{1}+ x{2}+ \cdots + x{n}}{n}Q{n} \\ &=\sqrt{\sum \limits{i=1}^{n}x{i}^{2}}= \sqrt{\frac{x{1}^{2}+ x{2}^{2}+ \cdots + x{n}^{2}}{n}}\end{aligned}$$

$$H{n}\leq G{n}\leq A{n}\leq Q{n}$$