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}}} .
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