PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (192)
Orphanage
Unclass'd
Unproven (496)
Corrections (20)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: Very high
[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.



"comparison of Pythagorean means" is owned by pahio.
(view preamble)

View style:

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

This object's parent.
Log in to rate this entry.
(view current ratings)

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.
Accessed 495 times total.

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)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy
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
[ reply | up ]
Interact
post | correct | update request | add example | add (any)