|
|
|
Viewing Version
1
of
'comparison of Pythagorean means'
|
[ view 'comparison of Pythagorean means'
|
back to history
]
| Title of object: |
comparison of Pythagorean means |
| Canonical Name: |
ComparisonOfPythagoreanMeans |
| Type: |
Topic |
| Created on: |
2008-02-16 10:54:55 |
| Modified on: |
2008-02-16 10:54:55 |
| Classification: |
msc:11-00, msc:62-07 |
| Defines: |
Pythagorean means |
Revision comment (for changes between this and next version):
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
\usepackage{pstricks}
\usepackage{pst-plot}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
|
Content:
If $u$ and $v$ are positive numbers and $u \leqq v$, then their {\em 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 \,$c(u,c)$, obey the \PMlinkname{order}{TotalOrder}
$$ u \leqq h(u,v) \leqq g(u,v) \leqq a(u,v) \leqq c(u,v) \leqq v.\\$$
The below plots the means\, $h(x,1)$ in black,\, $g(x,1)$ in \PMlinkescapetext{blue},\, $a(x,1)$ in cyan and\, $c(x,1)$ in green for\, $0 \leqq x \leqq 1$.
\begin{center}
\psset{unit=5cm}
\begin{pspicture}(-0.2,-0.2)(1.2,1.2)
\psaxes[Dx=1,Dy=1]{->}(0,0)(-0.1,-0.1)(1.1,1.1)
\rput(-0.03,-0.05){$0$}
\psdot(1,1)
\psline[linestyle=dotted](1,0)(1,1)
\psline[linestyle=dotted](0,1)(1,1)
\psplot[linecolor=green]{0}{1}{x x mul 1 add x 1 add div}
\psplot[linecolor=cyan]{0}{1}{x 1 add 2 div}
\psplot[linecolor=blue]{0}{1}{x sqrt}
\psplot[linecolor=black]{0}{1}{2 x mul x 1 add div}
\end{pspicture}
\end{center}
Note, that the \PMlinkname{linear graph}{Slope} of the arithmetic mean is the common \PMlinkname{tangent}{TangentLine} all those curves in the point \,$(1,1)$, since here the derivatives of all functions have the value $\frac{1}{2}$.
|
|
|
|
|
|
