The harmonic mean 
of 
positive
numbers 
(where 
,
..., 
)
is the number 
defined by
 |
(1)
|
The harmonic mean of a list of numbers may be computed in the Wolfram Language using HarmonicMean[list].
The special cases of 
and 
are therefore given by
 |  |  |
(2)
|
 |  |  |
(3)
|
and so on.
The harmonic means of the integers from 1 to 
for 
, 2, ... are 1, 4/3, 18/11, 48/25, 300/137, 120/49, 980/363,
... (OEIS A102928 and A175441).
For 
,
the harmonic mean is related to the arithmetic mean 
and geometric
mean 
by
 |
(4)
|
(Havil 2003, p. 120).
The harmonic mean is the special case 
of the power mean and is
one of the Pythagorean means. In older literature,
it is sometimes called the subcontrary mean.
The volume-to-surface area ratio for a cylindrical container with height 
and radius 
and the mean curvature of
a general surface are related to the harmonic mean.
Hoehn and Niven (1985) show that
 |
(5)
|
for positive numbers 
that are not all equal and any positive
constant 
.
Equality holds when all the 
are equal.
