# Lehmer mean

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

$L_{p}(x)={\frac {\sum _{k=1}^{n}x_{k}^{p}}{\sum _{k=1}^{n}x_{k}^{p-1}}}$ .

The Weighted Lehmer mean with respect to a tuple $w$ of positive weights is defined as:

$L_{p,w}(x)={\frac {\sum _{k=1}^{n}w_{k}\cdot x_{k}^{p}}{\sum _{k=1}^{n}w_{k}\cdot x_{k}^{p-1}}}$ .

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 $p\mapsto L_{p}(x)$ is non-negative

${\frac {\partial }{\partial p}}L_{p}(x)={\frac {\sum _{j=1}^{n}\sum _{k=j+1}^{n}(x_{j}-x_{k})\cdot (\ln x_{j}-\ln x_{k})\cdot (x_{j}\cdot x_{k})^{p-1}}{\left(\sum _{k=1}^{n}x_{k}^{p-1}\right)^{2}}},$ thus this function is monotonic and the inequality

$p\leq q\Rightarrow L_{p}(x)\leq L_{q}(x)$ holds.

## Special cases

• $\lim _{p\to -\infty }L_{p}(x)$ is the minimum of the elements of $x$ .
• $L_{0}(x)$ is the harmonic mean.
• $L_{\frac {1}{2}}(x)$ is the geometric mean.
• $L_{1}(x)$ is the arithmetic mean.
• $L_{2}(x)$ is the contraharmonic mean.
• $\lim _{p\to \infty }L_{p}(x)$ is the maximum of the elements of $x$ .
Sketch of a proof: Let $x_{1},\dots ,x_{k}$ be the values which equal the maximum. Then $L_{p}(x)=x_{0}\cdot {\frac {k+\left({\frac {x_{k+1}}{x_{0}}}\right)^{p}+\dots +\left({\frac {x_{n}}{x_{0}}}\right)^{p}}{k+\left({\frac {x_{k+1}}{x_{0}}}\right)^{p-1}+\dots +\left({\frac {x_{n}}{x_{0}}}\right)^{p-1}}}$ ## 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 $p$ and emphasizes big signal values for big $p$ . Given an efficient implementation of a moving arithmetic mean called smooth you can implement a moving Lehmer mean according to the following Haskell code.

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

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