Power mean inequality

Case $p>0$. Consider $b_i=a_i^p$ and $r=q/p>1$. The function $x\mapsto x^r$ is convex, so by Jensen:
$$\left(\frac1n\sum b_i\right)^r\le\frac1n\sum b_i^r.$$
That is, $M_p^{pr}\le M_q^q$, so $M_p^p\le M_q^q\cdot M_p^{p(q/p-1)}$... simplifying: $M_p\le M_q$.

Case $p=0$. This is AM-GM (geometric mean $\le$ arithmetic mean).

Case $p<0$. Similar, using convexity of $x\mapsto x^{q/p}$ for $q/p<1$ and $x>0$ (concave); the inequality reverses appropriately.

Conclusion. $M_p$ is increasing in $p$, and for any $p<q$, $M_p\le M_q$.

Example. $M_{-1}$ (harmonic) $\le M_0$ (geometric) $\le M_1$ (arithmetic) $\le M_2$ (quadratic).