Harmonic Mean

The harmonic mean $H(x_1, \ldots, x_n)$ of $n$ points $x_i$ (where $i=1$, ..., $n$) is

$${1\over H}\equiv {1\over n} \sum_{i=1}^n {1\over x_i}.$$ (1)

The special case of $n=2$ is therefore

$${1\over H}={1\over 2}\left({{1\over x_1}+{1\over x_2}},\right)$$ (2)

or

$${1\over H}={x_1+x_2\over 2x_1x_2}.$$ (3)

The Volume-to-Surface Area ratio for a cylindrical container with height $h$ and radius $r$ and the Mean Curvature of a general surface are related to the harmonic mean.

Hoehn and Niven (1985) show that

$$H(a_1+c, a_2+c, \ldots, a_n+c)>c+H(a_1, a_2, \ldots, a_n)$$ (4)

for any Positive constant $c$.

See also Arithmetic Mean, Arithmetic-Geometric Mean, Geometric Mean, Harmonic-Geometric Mean, Root-Mean-Square

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.

Hoehn, L. and Niven, I. ``Averages on the Move.'' Math. Mag. 58, 151-156, 1985.