How many other means exist out there? How do they relate to each other?

Question

I am aware of the Quadratic Mean, Arithmetic Mean, Geometric Mean, and Harmonic Mean, and they are related by the inequality chain $QM \ge AM \ge GM \ge HM$. For two numbers $a,b$, the quadratic mean is equal to $\sqrt{\frac{a^2+b^2}2}$, the arithmetic mean equals $\frac{a+b}2$, the geometric mean equals $\sqrt{ab}$, the harmonic mean equals $\frac{2}{\frac1a+\frac1b}$. Some means I made up: Logarithmic mean ($ln(e^a+e^b)$) and Trigonometric Mean: $arcsin(sin(a)+sin(b))$. So, are their any other such means?

Answer 1

$M_r(a)=(\frac{1}{n}\sum a^r)^{\frac{1}{r}}$ gives a whole class of means. In the formula $a$ stands for a sequence of numbers you are taking the mean of, and $r$ is the 'order' if you like, of the mean. So $$\lim_{r\rightarrow \infty}M_r(a)=\max a$$ and $M_{-\infty}=\min a$. The less trivial connection is a connection with the geometric means$$\lim_{r\rightarrow0}M_r(a)=G(a)$$ Moreover the means can be generalized a bit to have different weights $$M_r(a)=(\sum \omega a^r)^{\frac{1}{r}}$$ where the $\omega$ are non-negative and $\sum \omega=1$.

However you define the mean, it needs to satisfy homogeneity $M(ka)=kM(a)$ and $$\min a\leq M(a)\leq\max a$$ these are like the 'axioms' for means and are similar to those of group theory, topological space etc.