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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$.

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

Also defines:  Pythagorean means, Babylonian inequality chain

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Cross-references: Heronian mean, graph, functions, derivatives, point, curves, contraharmonic mean, arithmetic mean, geometric mean, harmonic mean, viz, numbers, positive

This is version 8 of comparison of Pythagorean means, born on 2008-02-16, modified 2008-02-22.
Object id is 10280, canonical name is ComparisonOfPythagoreanMeans.
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Classification:
AMS MSC: 01A17 (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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