# Lehmer mean

In mathematics, the **Lehmer mean** of a tuple of positive real numbers, named after Derrick Henry Lehmer,^{[1]} 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.

## 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 of the two values and .
- is the arithmetic mean.
- is the contraharmonic mean.
- is the maximum of the elements of .

- Sketch of a proof: Without loss of generality 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.

## See also

## Notes

- ↑ P. S. Bullen.
*Handbook of means and their inequalities*. Springer, 1987.