A post on Monday looked at means an inequalities for a lists of non-negative numbers. This post looks at analogous means and inequalities for non-negative *functions*. We go from means defined in terms of sums to means defined in terms of integrals.

Let *p*(*x*) be a non-negative function that integrates to 1. (That makes *p*(*x*) a probability density, but you don’t have to think of this in terms of probability.) Think of *p*(*x*) as being a weighting function. The dependence on *p*(*x*) will be implicit. For a non-negative function *f*(*x*) and real number *r* ≠ 0, define *M _{r}*(

*f*) by

If *r* > 0 and the integral defining *M _{r}*(

*f*) diverges, we say

*M*(

_{r}*f*) = ∞ and

*M*

_{–r}(

*f*) = 0.

Define the geometric mean of the function *f* by

There are a few special cases to consider when defining *G*(*f*). See Inequalities for details.

First we give several limiting theorems.

- As
*r*→ –∞,*M*(_{r}*f*) → min(*f*) - As
*r*→ +∞,*M*(_{r}*f*) → max(*f*) - As
*r*→ 0^{+},*M*(_{r}*f*) →*G*(*f*)

And now for the big theorem: If *r* ≤ *s*, then *M _{r}*(

*f*) ≤

*M*(

_{s}*f*).The conditions under which equality hold are a little complicated. Again, see Inequalities for details.

We could derive analogous results for infinite sums since sums are just a special case of integrals.

The assumption that the weight function *p*(*x*) has a finite integral is critical. We could change the definition of *M*_{r}( *f* ) slightly to accommodate the case that the integral of *p*(*x*) is finite but not equal to 1, and all the conclusions above would remain true. But if we allowed *p*(*x*) to have a divergent interval, the theorems do not hold. Suppose *p*(*x*) is constantly 1, and our region of integration is (0, ∞). Then *M*_{r}( *f* ) might be more or less than *M _{s}*(

*f*) depending on

*f*. For example, let

*f*(

*x*) =

*b*exp( –

*bx*) for some

*b*> 0.

*M*

_{1}(

*f*) = 1, but

*M*

_{∞}(

*f*) =

*b*. Then

*M*

_{1}(

*f*) is less than or greater than

*M*

_{∞}(

*f*) depending on whether

*b*is less than or greater than 1.