Lehmer mean

Let p be a real number.  Lehmer meanMathworldPlanetmath of the positive numbers a1,,an is defined as

Lp(a1,,an):=a1p++anpa1p-1++anp-1. (1)

This definition fulfils both requirements set for means (http://planetmath.org/Mean3).  In the case of Lehmer mean of two positive numbers a and b we see for  ab  that


The Lehmer mean of certain numbers is the greater the greater is the parametre p, i.e.


This turns out from the nonnegativeness of the partial derivativeMathworldPlanetmath of Lp with respect to p; in the case  n=2  it writes

Lpp=ap-1bp-1(a-b)(lna-lnb)(ap-1+bp-1)2 0.

Thus in the below list containing special cases of Lehmer mean, the is the least and the contraharmonic the greatest (cf. the comparison of Pythagorean means).

E.g. for two arguments a and b, one has

Note.  The least (http://planetmath.org/LeastNumber) and the greatest of the numbers (http://planetmath.org/GreatestNumber) a1,,an may be regarded as borderline cases of the Lehmer mean, since


For proving these equations, suppose that there are exactly k greatest (resp. least) ones among the numbers and that those are  a1==ak.  Then we can write


Letting  p+  (resp. p-),  this equation yields

Title Lehmer mean
Canonical name LehmerMean
Date of creation 2013-03-22 19:02:06
Last modified on 2013-03-22 19:02:06
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Definition
Classification msc 62-07
Classification msc 11-00
Related topic OrderOfSixMeans
Related topic LeastAndGreatestNumber
Related topic MinimalAndMaximalNumber