Average
with weight ,
,
, of a set of real numbers
A variable
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where is a continuous strictly-monotone function on
. When
, one obtains
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and, in particular, when ,
,
will be the arithmetic average of the numbers
, while when
, it will be the harmonic average. The concepts of the geometric average
and the weighted geometric average
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are introduced separately.
One of the basic results of the theory of averages is the inequality , except when all
are equal to each other. Other results are:
1) ;
2) if and only if
,
,
;
3) if and only if
is a convex function; in particular
if
.
The concept of an average can be extended to infinite sequences under the assumption that the corresponding series and products converge, and to other functions. The following is such an example:
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given the condition that almost everywhere on the corresponding interval and that
. Thus,
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References
[1] | G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934) |
Comments
Instead of "average" the term "meanmean" is also quite often used: arithmetic mean, geometric mean, etc.
References
[a1] | D.S. Mitrinović, "Analytic inequalities" , Springer (1970) |
[a2] | D.S. Mitrinović, "Elementary inequalities" , Noordhoff (1964) |
Average. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Average&oldid=33369