Stolarsky means

In mathematics is Stolarskysche average or short the Stolarsky means a term introduced by Kenneth B. Stolarsky mean that the logarithmic means generalized.

For two numbers and one parameter , the Stolarsky mean is defined as ${\displaystyle x,y}$ ${\displaystyle p}$

${\displaystyle S_{p}(x,y)\,=\,\lim _{(\xi ,\eta )\to (x,y)}\left({\frac {\xi ^{p}-\eta ^{p}}{p(\xi -\eta )}}\right)^{1 \over p-1}\,=\,{\begin{cases}x&{\mbox{falls }}x=y\\\left({\frac {x^{p}-y^{p}}{p(x-y)}}\right)^{1 \over p-1}&{\mbox{sonst}}\end{cases}}}$

In this case the limit is over all pairs with form to. In this case , the limit value is the -th power of the differential quotient of the function and therefore actually agrees with, as indicated . ${\displaystyle (\xi ,\eta )}$ ${\displaystyle \xi \not =\eta }$ ${\displaystyle x=y}$ ${\displaystyle {1 \over p-1}}$ ${\displaystyle x\mapsto {\frac {x^{p}}{p}}}$ ${\displaystyle x}$

Special cases

The Stolarsky remedy has the following special cases:

${\displaystyle S_{-\infty }(x,y)}$	${\displaystyle \,={\text{min}}\{x,y\}}$	Minimum (limit!)
${\displaystyle \,S_{-1}(x,y)}$	${\displaystyle ={\sqrt {xy}}}$	Geometric mean
${\displaystyle \,S_{0}(x,y)}$	${\displaystyle ={\frac {x-y}{\log x-\log y}}}$	Logarithmic mean (limit value!)
${\displaystyle S_{\frac {1}{2}}(x,y)}$	${\displaystyle =\left({\frac {{\sqrt {x}}+{\sqrt {y}}}{2}}\right)^{2}}$	Holder means with 1/2
${\displaystyle \,S_{1}(x,y)}$	${\displaystyle ={\frac {1}{e}}\left({\frac {y^{y}}{x^{x}}}\right)^{\frac {1}{y-x}}}$	identric mean (limit value!)
${\displaystyle \,S_{2}(x,y)}$	${\displaystyle ={\frac {x+y}{2}}}$	Arithmetic mean
${\displaystyle S_{\infty }(x,y)}$	${\displaystyle =\,{\text{max}}\{x,y\}}$	Maximum (limit value!)

Weighted Stolarsky mean

The Stolarsky mean can also be weighted:

${\displaystyle S_{\omega _{1},\omega _{2}}(x,y)=\left({\frac {\omega _{2}\cdot (x^{\omega _{1}}-y^{\omega _{1}})}{\omega _{1}\cdot (x^{\omega _{2}}-y^{\omega _{2}})}}\right)^{\frac {1}{\omega _{1}-\omega _{2}}}}$

credentials

• Horst Alzer: Best possible estimates for special mean values . (PDF; 141 kB)
• Horst Alzer: Inequalities for mean values . In: Archives of Mathematics , Vol. 47 (5), November 1986, springerlink.com
• Edward Neumann: Stolarski Means of Several Variables (PDF; 254 kB) In: Journal of Inequalities in Pure and Applied Mathematics , Vol. 6, 2 (30), 2005.
• Thomas Riedel, Prasanna K. Sahoo: A characterization of the Stolarsky mean . In: Aequationes Mathematicae , 70, No. 1/2, Sept. 2005, springerlink.com

Individual evidence

1. ↑ Kenneth B. Stolarsky: Generalizations of the logarithmic mean . In: Mathematics Magazine , Vol. 48, No. March 2, 1975, pp. 87-92
2. ↑ Eric W. Weisstein : Stolarsky mean . In: MathWorld (English).
3. ↑ Julian Havil: Gamma: Euler's constant, prime number beaches and the Riemann hypothesis . Springer, Berlin 2007, ISBN 978-3-540-48495-0
4. ↑ Eric W. Weisstein : Identric Mean . In: MathWorld (English).
5. ↑ Laszlo Losonczi: Ratio of Stolarsky means: monotonicity and comparison