Means and inequalities

by John

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 = (x₁, x₂, x₃…, 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 rdecreases 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 M₁, M₀, 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 rgoes to –∞. Then M_∞( x ) equals max(x₁, x₂, x₃…, x_n) and M_-∞( x ) equals min(x₁, x₂, x₃…, 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 ) <>s( x ) unless x₁ = x₂ = x₃ = … = 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 p₁ + p₂ + p₃ + … + 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.