In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.

Definition

For two positive real numbers x {\displaystyle x} and y {\displaystyle y} the Stolarsky Mean is defined as:

S p ( x , y ) = { x , if x = y , ( x p − y p p ( x − y ) ) 1 / ( p − 1 ) , otherwise . {\displaystyle S_{p}(x,y)=\left\{{\begin{array}{l l}x,&{\text{if }}x=y,\\\left({\frac {x^{p}-y^{p}}{p(x-y)}}\right)^{1/(p-1)},&{\text{otherwise}}.\end{array}}\right.}

Derivation

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 =\left[f'\right]^{-1}\left({\frac {f(x)-f(y)}{x-y}}\right)}

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

Special cases

  • lim p → − ∞ S p ( x , y ) {\displaystyle \lim _{p\to -\infty }S_{p}(x,y)} is the minimum.
  • S − 1 ( x , y ) {\displaystyle S_{-1}(x,y)} is the geometric mean.
  • lim p → 0 S p ( x , y ) {\displaystyle \lim _{p\to 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}.
  • S 1 2 ( x , y ) {\displaystyle S_{\frac {1}{2}}(x,y)} is the power mean with exponent 1 2 {\displaystyle {\frac {1}{2}}}.
  • lim p → 1 S p ( x , y ) {\displaystyle \lim _{p\to 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)=x\cdot \ln x}.
  • S 2 ( x , y ) {\displaystyle S_{2}(x,y)} is the arithmetic mean.
  • S 3 ( x , y ) = Q M ( x , y , G M ( x , y ) ) {\displaystyle S_{3}(x,y)=QM(x,y,GM(x,y))} is a connection to the quadratic mean and the geometric mean.
  • lim p → ∞ S p ( x , y ) {\displaystyle \lim _{p\to \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

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

See also