Stolarsky mean

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

DefinitionEdit

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

 

DerivationEdit

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 casesEdit

  •   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.

GeneralizationsEdit

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 alsoEdit

ReferencesEdit

  1. ^ 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.