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| In [[mathematics]], the '''Stolarsky mean''' of two positive [[real number]]s ''x'', ''y'' is defined as:
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| :<math>
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| \begin{align}
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| S_p(x,y)
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| & = \lim_{(\xi,\eta)\to(x,y)}
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| \left({\frac{\xi^p-\eta^p}{p (\xi-\eta)}}\right)^{1/(p-1)} \\[10pt]
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| & = \begin{cases}
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| x & \text{if }x=y \\
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| \left({\frac{x^p-y^p}{p (x-y)}}\right)^{1/(p-1)} & \text{else}
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| \end{cases}
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| \end{align}
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| </math>
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| It is derived from the [[mean value theorem]], which states that a [[secant line]], cutting the graph of a [[differentiable]] function <math>f</math> at <math>( x, f(x) )</math> and <math>( y, f(y) )</math>, has the same [[slope]] as a line [[tangent]] to the graph at some point <math>\xi</math> in the [[Interval (mathematics)|interval]] <math>[x,y]</math>.
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| :<math> \exists \xi\in[x,y]\ f'(\xi) = \frac{f(x)-f(y)}{x-y} </math>
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| The Stolarsky mean is obtained by | |
| :<math> \xi = f'^{-1}\left(\frac{f(x)-f(y)}{x-y}\right) </math>
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| when choosing <math>f(x) = x^p</math>.
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| == Special cases ==
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| *<math>\lim_{p\to -\infty} S_p(x,y)</math> is the [[minimum]].
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| *<math>S_{-1}(x,y)</math> is the [[geometric mean]].
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| *<math>\lim_{p\to 0} S_p(x,y)</math> is the [[logarithmic mean]]. It can be obtained from the mean value theorem by choosing <math>f(x) = \ln x</math>.
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| *<math>S_{\frac{1}{2}}(x,y)</math> is the [[power mean]] with exponent <math>\frac{1}{2}</math>.
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| *<math>\lim_{p\to 1} S_p(x,y)</math> is the [[identric mean]]. It can be obtained from the mean value theorem by choosing <math>f(x) = x\cdot \ln x</math>.
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| *<math>S_2(x,y)</math> is the [[arithmetic mean]].
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| *<math>S_3(x,y) = QM(x,y,GM(x,y))</math> is a connection to the [[quadratic mean]] and the [[geometric mean]].
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| *<math>\lim_{p\to\infty} S_p(x,y)</math> is the [[maximum]].
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| == Generalizations ==
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| You can generalize the mean to ''n'' + 1 variables by considering the [[mean value theorem for divided differences]] for the ''n''th [[derivative]].
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| You obtain
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| :<math>S_p(x_0,\dots,x_n) = {f^{(n)}}^{-1}(n!\cdot f[x_0,\dots,x_n])</math> for <math>f(x)=x^p</math>.
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| == See also ==
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| *[[mean]]
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| == References ==
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| {{reflist}}
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| * {{cite journal | zbl=0302.26003 | last=Stolarsky | first=Kenneth B. | title=Generalizations of the logarithmic mean | journal=[[Mathematics Magazine]] | volume=48 | pages=87–92 | year=1975 | issn=0025-570X | url=http://links.jstor.org/sici?sici=0025-570X%28197503%2948%3A2%3C87%3AGOTLM%3E2.0.CO%3B2-6 }}
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| [[Category:Means]]
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Friends contact her Claude Gulledge. The preferred hobby for my kids and me is dancing and now I'm trying to make cash with it. Bookkeeping is what I do for a living. Years in the past we moved to Kansas.
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