Stolarsky mean
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 obtains