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Stolarsky mean

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In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.[1]

Definition

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For two positive real numbers xy the Stolarsky Mean is defined as:

Derivation

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

The Stolarsky mean is obtained by

when choosing .

Special cases

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  • is the minimum.
  • is the geometric mean.
  • is the logarithmic mean. It can be obtained from the mean value theorem by choosing .
  • is the power mean with exponent .
  • is the identric mean. It can be obtained from the mean value theorem by choosing .
  • is the arithmetic mean.
  • is a connection to the quadratic mean and the geometric mean.
  • is the maximum.

Generalizations

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

for .

See also

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References

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  1. ^ Stolarsky, Kenneth B. (1975). "Generalizations of the logarithmic mean". Mathematics Magazine. 48 (2): 87–92. doi:10.2307/2689825. ISSN 0025-570X. JSTOR 2689825. Zbl 0302.26003.