INEQUALITIES: Arithmetic mean, Geometric mean, Harmonic mean and Root mean square

Let 

$H\le G\le A\le S$

where:

• $A={\displaystyle \frac{a_1+a_2+\cdots +a_n}{n}}$ - arithmetic mean
• $G=\sqrt[n]{a_1{a}_2\cdots a_n}$ - geometric mean
• $H={\displaystyle \frac{n}{\frac{1}{a_1}+{\displaystyle \frac{1}{a_2}}+\cdots +{\displaystyle \frac{1}{a_n}}}}$ - harmonic mean
• $S=\sqrt{{\displaystyle \frac{a_1^2+a_2^2+\cdots +a_n^2}{n}}}$ - root mean square

If $a_1=a_2=\cdots =a_n$, then 

$H=G=A=S$.