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

Let 

$H \leq G \leq A \leq S$

where:

• $A = \dfrac{a_1 + a_2 + \dotsb + a_n}{n}$ - arithmetic mean
• $G = \sqrt[n]{a_1a_2\dotsb a_n}$ - geometric mean
• $H = \dfrac{n}{\frac{1}{a_1} + \dfrac{1}{a_2} + \dots + \dfrac{1}{a_n}}$ - harmonic mean
• $S = \sqrt{\dfrac{a_1^2 + a_2^2 + \dots + a_n^2}{n}}$ - root mean square

If $a_1 = a_2 = \dots = a_n$, then 

$H = G = A = S$.