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[parent] comparison of Pythagorean means (Topic)

If $u$ and $v$ are positive numbers and $u \leqq v$ , 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

$\displaystyle u \leqq h(u,v) \leqq g(u,v) \leqq a(u,v) \leqq c(u,v) \leqq v.$ (1)

The part
$\displaystyle u \leqq h(u,v) \leqq g(u,v) \leqq a(u,v) \leqq v$ (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 $0 \leqq x \leqq 1$ .


\begin{pspicture}(-1,-1)(5.9,5.9) \psaxes[Dx=10,Dy=10]{->}(0,0)(-0.5,-0.5)(5.7,5... ... \psplot[linecolor=yellow]{0}{5}{x 5 x mul sqrt add 5 add 3 div} \end{pspicture}

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.

Bibliography

1
HORST HISCHER: ``Viertausend Jahre Mittelwertbildung''. -- mathematica didactica 25 (2002). See also this.




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See Also: arithmetic-geometric-harmonic means inequality, general means inequality, Heronian mean is between geometric and arithmetic mean, integer contraharmonic means, order of six means

Also defines:  Pythagorean means, Babylonian inequality chain

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Cross-references: quadratic mean, Heronian mean, graph, functions, derivatives, point, curves, diagram, contraharmonic mean, arithmetic mean, geometric mean, harmonic mean, viz, numbers, positive
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This is version 9 of comparison of Pythagorean means, born on 2008-02-16, modified 2009-01-22.
Object id is 10280, canonical name is ComparisonOfPythagoreanMeans.
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Classification:
AMS MSC01A17 (History and biography :: History of mathematics and mathematicians :: Babylonian)
 01A20 (History and biography :: History of mathematics and mathematicians :: Greek, Roman)
 11-00 (Number theory :: General reference works )
 62-07 (Statistics :: Data analysis)

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no graph seen in HTML mode by pahio on 2008-02-17 09:51:42
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
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