Lehmer mean

In mathematics, the Lehmer mean of a tuple ${\displaystyle x}$ of positive real numbers, named after Derrick Henry Lehmer,^[1] is defined as:

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

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

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

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