Lehmer mean

In mathematics, the Lehmer mean of a tuple ${\displaystyle x}$ of positive real numbers, named after Derrick Henry Lehmer, is defined as:

The weighted Lehmer mean with respect to a tuple ${\displaystyle w}$ of positive weights is defined as:

The Lehmer mean is an alternative to power means for interpolating between minimum and maximum via arithmetic mean and harmonic mean.

The derivative of ${\displaystyle p\mapsto L_{p}(\mathbf {x} )}$ is non-negative

thus this function is monotonic and the inequality

holds.

The derivative of the weighted Lehmer mean is:

Like a power mean, a Lehmer mean serves a non-linear moving average which is shifted towards small signal values for small ${\displaystyle p}$ and emphasizes big signal values for big ${\displaystyle p}$. Given an efficient implementation of a moving arithmetic mean called smooth you can implement a moving Lehmer mean according to the following Haskell code.