Means and inequalities for functions

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 Mr( f ) by

M_r(f) = \left(\int p(x) \left( f(x) \right)^r \, dx \right)^{1/r}

If r > 0 and the integral defining Mr( f ) diverges, we say Mr( f ) = ∞ and Mr( f ) = 0.

Define the geometric mean of the function f by

G(f) = \exp\left( \int p(x) \log f(x)\, dx \right)

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

First we give several limiting theorems.

  • As r → –∞, Mr( f ) → min( f )
  • As r → +∞, Mr( f ) → max( f )
  • As r → 0+, Mr( f ) → G( f )

And now for the big theorem: If rs, then Mr( f ) ≤ Ms( 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 Mr( 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 Mr( f ) might be more or less than Ms( f ) depending on f. For example, let f(x) = b exp( − bx ) for some b > 0. M1( f ) = 1, but M( f ) = b. Then M1( f ) is less than or greater than M( f ) depending on whether b is less than or greater than 1.

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