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\le q \Rightarrow L_p(x) \le 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 $$L_p(x)=x_0\cdot\frac{k+\left(\frac{x_{k+1}}{x_0}\right)^p+\dots+\left(\frac{x_{n}}{x_0}\right)^p}{k+\left(\frac{x_{k+1}}{x_0}\right)^{p-1}+\dots+\left(\frac{x_{n}}{x_0}\right)^{p-1}}$$

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.