Clay means

In mathematics , the Lehmer mean is a generalized mean named after Derrick Henry Lehmer .

definition

The Lehmer mean of positive real numbers for the level is defined as follows: ${\displaystyle n}$ ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle p}$

${\displaystyle L_{p}(a_{1},\ldots ,a_{n})={\frac {\sum _{k=1}^{n}a_{k}^{p}}{\sum _{k=1}^{n}a_{k}^{p-1}}}.}$

There is also a form of the clay remedy with (positive) weights . The weighted Lehmer mean is: ${\displaystyle w=(w_{1},\ldots ,w_{n})}$

${\displaystyle L_{p,w}(a_{1},\ldots ,a_{n})={\frac {\sum _{k=1}^{n}w_{k}a_{k}^{p}}{\sum _{k=1}^{n}w_{k}a_{k}^{p-1}}}.}$

properties

The following applies to the Lehmer remedy

• ${\displaystyle \lim _{p\to -\infty }L_{p}(a_{1},\ldots ,a_{n})=\min\{a_{1},\ldots ,a_{n}\}}$ is the minimum value.
• ${\displaystyle L_{0}(a_{1},\ldots ,a_{n})={\frac {n}{\sum _{k=1}^{n}{\frac {1}{a_{k}}}}}={\frac {n}{{\frac {1}{a_{1}}}+\cdots +{\frac {1}{a_{n}}}}}}$is the harmonic mean .
• For is the geometric mean . ${\displaystyle n=2}$ ${\displaystyle L_{\frac {1}{2}}(a_{1},a_{2})={\frac {{\sqrt {a_{1}}}+{\sqrt {a_{2}}}}{{\frac {1}{\sqrt {a_{1}}}}+{\frac {1}{\sqrt {a_{2}}}}}}={\sqrt {a_{1}a_{2}}}}$
• ${\displaystyle L_{1}(a_{1},\ldots ,a_{n})={\frac {\sum _{k=1}^{n}a_{k}}{n}}={\frac {a_{1}+\cdots +a_{n}}{n}}}$is the arithmetic mean .
• ${\displaystyle L_{2}(a_{1},\ldots ,a_{n})={\frac {\sum _{k=1}^{n}a_{k}^{2}}{a_{1}+\cdots +a_{n}}}}$is the contrarian means already known to Eudoxus by Knidos .
• ${\displaystyle \lim _{p\to \infty }L_{p}(a_{1},\ldots ,a_{n})=\max\{a_{1},\ldots ,a_{n}\}}$ is the maximum value.

In contrast to the other five special cases, the contrarian means is not monotonic, ie from for all does not follow . ${\displaystyle a_{i}\leq b_{i}}$ ${\displaystyle i}$ ${\displaystyle L_{2}(a_{1},\ldots ,a_{n})\leq L_{2}(b_{1},\ldots ,b_{n})}$

Individual evidence

1. ↑ Hischer / Lambert: "What is a numerical mean?"
2. ↑ Axioms of mean values

literature

• DH Lehmer: On the compounding of certain means. J. Math. Anal. Appl. 36 (1971) pp. 183-200
• P. S. Bullen: Handbook of Means and Their Inequalities . Kluwer Acad. Pub. 2003, ISBN 1-4020-1522-4 (comprehensive discussion of mean values and the inequalities associated with them).