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# comparison of Pythagorean means

If $u$ and $v$ are positive numbers and $u\leq v$, then their Pythagorean means, viz. the harmonic mean $h(u,v)$, the geometric mean $g(u,v)$, the arithmetic mean $a(u,v)$ and the contraharmonic mean $c(u,v)$, obey the order

$\displaystyle u\;\leq\;h(u,v)\;\leq\;g(u,v)\;\leq\;a(u,v)\;\leq\;c(u,v)\;\leq% \;v.$ | (1) |

The part

$\displaystyle u\;\leq\;h(u,v)\;\leq\;g(u,v)\;\leq\;a(u,v)\;\leq\;v$ | (2) |

of (1) was known already by the ancient Babylonians. Therefore it may be called the Babylonian inequality chain (Horst Hischer).

The below diagram plots the means $h(x,1)$ in black, $g(x,1)$ in blue, $a(x,1)$ in cyan and $c(x,1)$ in green for $0\leq x\leq 1$.

Note that the linear graph of the arithmetic mean is the common tangent all those curves in the point $(1,1)$, since here the derivatives of all functions have the value $\frac{1}{2}$. The same concerns the yellow graph of the Heronian mean of $x$ and $1$, similarly the red graph of the quadratic mean.

# References

- 1 Horst Hischer: “Viertausend Jahre Mittelwertbildung”. — mathematica didactica 25 (2002). See also this.

## Mathematics Subject Classification

62-07*no label found*11-00

*no label found*01A20

*no label found*01A17

*no label found*

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## Comments

## no graph seen in HTML mode

Hi, the pstrics diagram of

http://planetmath.org/encyclopedia/ComparisonOfPythagoreanMeans.html

is seen well in page images mode, but not at all in HTML mode. Are there some experts who could help?

Jussi

## Re: no graph seen in HTML mode

I solved already the HTML problem (by leaving away "unit=5cm").