# Lehmer mean

The **Lehmer mean** of a tuple of positive real numbers is defined as:

- .

The **Weighted Lehmer mean** with respect to a tuple 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.

## Contents

## Properties

The derivative of is non-negative

thus this function is monotonic and the inequality

holds.

## Special cases

- is the minimum of the elements of .
- is the harmonic mean.
- is the geometric mean.
- is the arithmetic mean.
- is the contraharmonic mean.
- is the maximum of the elements of .

- Sketch of a proof: Let be the values which equal the maximum. Then

## Applications

### Signal processing

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

lehmerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a] lehmerSmooth smooth p xs = zipWith (/) (smooth (map (**p) xs)) (smooth (map (**(p-1)) xs))

- For big it can serve an envelope detector on a rectified signal.
- For small it can serve an baseline detector on a mass spectrum.