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In mathematics, the Stolarsky mean of two positive real numbers x, y is defined as:

Sp(x,y)=lim(ξ,η)→(x,y)(ξp−ηpp(ξ−η))1/(p−1)={xif x=y(xp−ypp(x−y))1/(p−1)else{displaystyle {begin{aligned}S_{p}(x,y)&=lim _{(xi ,eta )to (x,y)}left({frac {xi ^{p}-eta ^{p}}{p(xi -eta )}}right)^{1/(p-1)}\[10pt]&={begin{cases}x&{text{if }}x=y\left({frac {x^{p}-y^{p}}{p(x-y)}}right)^{1/(p-1)}&{text{else}}end{cases}}end{aligned}}}

It is derived from the mean value theorem, which states that a secant line, cutting the graph of a differentiable function f{displaystyle f} at (x,f(x)){displaystyle (x,f(x))} and (y,f(y)){displaystyle (y,f(y))}, has the same slope as a line tangent to the graph at some point ξ{displaystyle xi } in the interval [x,y]{displaystyle [x,y]}.

∃ξ∈[x,y] f′(ξ)=f(x)−f(y)x−y{displaystyle exists xi in [x,y] f'(xi )={frac {f(x)-f(y)}{x-y}}}

The Stolarsky mean is obtained by

ξ=f′−1(f(x)−f(y)x−y){displaystyle xi =f'^{-1}left({frac {f(x)-f(y)}{x-y}}right)}

when choosing f(x)=xp{displaystyle f(x)=x^{p}}.

Special cases

• 
  limp→−∞Sp(x,y){displaystyle lim _{pto -infty }S_{p}(x,y)} is the minimum.


• 
  S−1(x,y){displaystyle S_{-1}(x,y)} is the geometric mean.


• 
  limp→0Sp(x,y){displaystyle lim _{pto 0}S_{p}(x,y)} is the logarithmic mean. It can be obtained from the mean value theorem by choosing f(x)=ln⁡x{displaystyle f(x)=ln x}.


• 
  S12(x,y){displaystyle S_{frac {1}{2}}(x,y)} is the power mean with exponent 12{displaystyle {frac {1}{2}}}.


• 
  limp→1Sp(x,y){displaystyle lim _{pto 1}S_{p}(x,y)} is the identric mean. It can be obtained from the mean value theorem by choosing f(x)=x⋅ln⁡x{displaystyle f(x)=xcdot ln x}.


• 
  S2(x,y){displaystyle S_{2}(x,y)} is the arithmetic mean.


• 
  S3(x,y)=QM(x,y,GM(x,y)){displaystyle S_{3}(x,y)=QM(x,y,GM(x,y))} is a connection to the quadratic mean and the geometric mean.


• 
  limp→∞Sp(x,y){displaystyle lim _{pto infty }S_{p}(x,y)} is the maximum.

Generalizations

One can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the nth derivative.
One obtains

Sp(x0,…,xn)=f(n)−1(n!⋅f[x0,…,xn]){displaystyle S_{p}(x_{0},dots ,x_{n})={f^{(n)}}^{-1}(n!cdot f[x_{0},dots ,x_{n}])} for f(x)=xp{displaystyle f(x)=x^{p}}.

See also

• Mean

References

• 
  Stolarsky, Kenneth B. (1975). "Generalizations of the logarithmic mean". Mathematics Magazine. 48: 87–92. doi:10.2307/2689825. ISSN 0025-570X. JSTOR 2689825. Zbl 0302.26003..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}