Do the Pythagorean means preserve the order among groups of observations?

Question

I know that

harmonic mean HM <= geometric mean GM <= arithmetic mean AM

Thus, if I have two groups of observations such that

AM of group 2 > AM of group 1

The smallest possible value of an observation is 1.

Is is true that the same inequality holds for the other two types of means:

GM of group 2 > GM of group 1 HM of group 2 > HM of group 1

Many thanks,

Simone

Answer 1

Absolutely no. You will still have $HM\leqslant GM\leqslant AM$ within each group, but that's about it.

Say, our groups are $\{20,20\}$ and $\{5,45\}$. Then $\stackrel{\color{red}{20}}{AM_1}\;<\;\stackrel{\color{red}{25}}{AM_2}$, but $\stackrel{\color{red}{20}}{GM_1}\;>\;\stackrel{\color{red}{15}}{GM_2}$ and $\stackrel{\color{red}{20}}{HM_1}\;>\;\stackrel{\color{red}{9}}{HM_2}$.