The **arithmetic** mean of two numbers *a* and *b* is (*a* + *b*)/2.

The **geometric** mean of *a* and *b* is √(*ab*).

The **harmonic** mean of *a* and *b* is 2/(^{1}/_{a} + ^{1}/_{b}).

This post will generalize these definitions of means and state a general inequality relating the generalized means.

Let *x* be a vector of non-negative real numbers, *x* = (*x*_{1}, *x*_{2}, *x*_{3}…, *x*_{n}). Define M_{r}( *x* ) to be

unless *r* = 0 or *r* is negative and one of the *x*_{i} is zero. If *r* = 0, define M_{r}( *x* ) to be the limit of M_{r}( *x* ) as *r* decreases to 0 . And if *r* is negative and one of the *x*_{i} is zero, define M_{r}( *x* ) to be zero. The arithmetic, geometric, and harmonic means correspond to M_{1}, M_{0}, and M_{-1} respectively.

Define M_{∞}( *x* ) to be the limit of M_{r}( *x* ) as *r* goes to ∞. Similarly, define M_{-∞}( *x* ) to be the limit of M_{r}( *x* ) as *r* goes to –∞. Then M_{∞}( *x* ) equals max(*x*_{1}, *x*_{2}, *x*_{3}…, *x*_{n}) and M_{-∞}( *x* ) equals min(*x*_{1}, *x*_{2}, *x*_{3}…, *x*_{n}).

In summary, the **minimum**, **harmonic mean**, **geometric mean**, **arithmetic mean** and **maximum **are all special cases of M_{r}( *x* ) corresponding to *r* = –∞, –1, 0, 1, and ∞ respectively. Of course other values of r are possible; these five are just the most familiar. Another common example is the root-mean-square (RMS) corresponding to *r* = 2.

A famous theorem says that the geometric mean is never greater than the arithmetic mean. This is a very special case of the following theorem.

If *r* ≤* s* then M_{r}( *x* ) ≤ M_{s}( *x* ).

In fact we can say a little more. If *r* <* s* then M_{r}( *x* ) < M_{s}( *x* ) unless *x*_{1} *= x*_{2} = *x*_{3 }= … = *x*_{n }or *s* ≤ 0 and one of the *x*_{i} is zero.

We could generalize the means M_{r} a bit more by introducing positive weights *p*_{i} such that *p*_{1} +* p*_{2} + *p*_{3} + … + *p*_{n} = 1. We could then define M_{r}( *x* ) as

with the same fine print as in the previous definition. The earlier definition reduces to this new definition with *p*_{i} = 1/*n*. The above statements about the means M_{r}( *x* ) continue to hold under this more general definition.

For more on means and inequalities, see Inequalities by Hardy, Littlewood, and Pólya.

**Update**: Analogous results for means of functions, replacing sums with integrals. Also, physical examples of harmonic mean with springs and resistors.

Related post: Old math books

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