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Stolarsky mean

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

Contents

S p ( x , y ) = lim ( ξ , η ) → ( x , y ) ( ξ p − η p p ( ξ − η ) ) 1 / ( p − 1 ) = { x if  x = y ( x p − y p p ( x − y ) ) 1 / ( p − 1 ) else

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

∃ ξ ∈ [ x , y ]   f ′ ( ξ ) = f ( x ) − f ( y ) x − y

The Stolarsky mean is obtained by

ξ = f ′ − 1 ( f ( x ) − f ( y ) x − y )

when choosing f ( x ) = x p .

Special cases

  • lim p → − ∞ S p ( x , y ) is the minimum.
  • S − 1 ( x , y ) is the geometric mean.
  • lim p → 0 S p ( x , y ) is the logarithmic mean. It can be obtained from the mean value theorem by choosing f ( x ) = ln ⁡ x .
  • S 1 2 ( x , y ) is the power mean with exponent 1 2 .
  • lim p → 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 .
  • S 2 ( x , y ) is the arithmetic mean.
  • S 3 ( x , y ) = Q M ( x , y , G M ( x , y ) ) is a connection to the quadratic mean and the geometric mean.
  • lim p → ∞ 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 ] ) for f ( x ) = x p .

    References

    Stolarsky mean Wikipedia
    (Text) CC BY-SA