the arithmetic, geometric and harmonic mean

For positive numbers a and b, the geometric mean is \sqrt{ab}.
e.g. If an investment earns 25% in the first year (i.e. the amount is multiplied by a=1.25) and 80% in the second year (b=1.8) then the average annual rate of return r is \sqrt{ab}=1.5, since ab=r^2.

For positive a,b,c\dots, \displaystyle\sqrt[n]{abc\dots}\les\frac{a+b+c+\dots}{n}
The two sides are equal when a=b=c=\dots
For n=2:

    \[\sqrt{ab}\les\frac{a+b}{2}\quad\text{(the AM-GM inequality)}\]

as shown in the following diagrams:

1.
amgm
i.e. when a\neq b, the rectangle \sqrt{a}\sqrt{b} is smaller than the area of the two triangular half-squares \displaystyle\frac{a}{2}+\frac{b}{2}

2.
AMGM 1
Why is the height h=\sqrt{ab}?
By similar triangles, \quad \displaystyle \frac{h}{a}=\frac{b}{h} \quad \to \quad h^2=ab \quad \to \quad h=\sqrt{ab}

3.
AMGM 2

4.
2 amgms

5.
amgm another

Harmonic mean

If we travel a distance d at rate r_1 in time t_1 and make the return trip at rate r_2 in time t_2, then r_1t_1=d=r_2t_2. What’s the average rate for the full trip?
2d=r(t_1+t_2) and therefore:

    \[r=\frac{2d}{t_1+t_2}=\frac{2d}{\frac{d}{r_1}+\frac{d}{r_2}}=\frac{2r_1r_2}{r_1+r_2}\]

This average rate is the harmonic mean of the two rates, and less than the average – the arithmetic mean \displaystyle\frac{r_1+r_2}{2} – when the rates differ, because more time is spent at the lower speed.

HM \les GM \les AM

i.e. \displaystyle\frac{2xy}{x+y}\les\sqrt{xy}\les\frac{x+y}{2}.

The video shows points in 0\les x,y \les 10.

Some other means

Root mean square (RMS) :+++ \displaystyle\sqrt{\frac{a^2+b^2}{2}}

Contraharmonic mean :+++ \displaystyle\frac{a^2+b^2}{a+b}

Heronian mean :+++ \displaystyle\frac{a+\sqrt{ab}+b}{3}

Logarithmic mean :+++ \displaystyle\frac{b-a}{\ln b - \ln a}

Identric mean :+++ \displaystyle\frac{(\frac{b^b}{a^a})^\frac{1}{b-a}}{e}

References
Alsina, Nelsen – When Less Is More
Nelsen – Proofs Without Words, I & II

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