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 and , the Stolarsky mean is defined as:
Derivation 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
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
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 Mean
References