At some point in our mid-teens, we breathed and ate the arithmetic-geometric mean inequality:
for any and any non-negative integers .
No doubt we pitied the poor souls who needed to have it stated. No other inequality could compare to it. In fact, we (meaning I) knew of nearly no other inequality: we saw the triangle inequality as common sense, knew it had an odd-looking higher-dimensional cousin called “Minkowski’s inequality”, and had seen Cauchy’s inequality once, but we’d never seen any of them applied in real life. (Admittedly, the triangle inequality gets used every time we walk straight to a place instead of taking the sides, but real life does not count as “real life” for the purposes of this paragraph.)
Then we did not hear anything about inequalities for a little while – and then Cauchy((-Bunyakovsky)-Schwarz) became ubiquitous. For non-mathematicians: Cauchy’s inequality states that for any real or complex numbers , or, verbally, “the dot product of two vectors is at most the square root of the product of their lengths”. A special case of it, namely,
is the Kalashnikov of analytic number theory: it can be used with ease by even a poorly trained student, with an impressively high kill ratio. In experienced hands, it is often one of several tools used to assault successfully an apparently impregnable target.
As soon as we learned some basic functional analysis, Cauchy and Minkowski fell into place, together with Hölder, of which Cauchy is a special case: Minkowski turned out to have stated that the expressions are bona fide norms even for , and Höder shows that the dual of such a norm is another one such norm. What is meant by “dual” here is an easily explainable notion with a great deal of depth –
– but we are getting farther away from our topic: what on earth happened to the arithmetic-geometric mean inequality? Only very, very rarely does it appear in actual mathematics. (To avoid a fuzzy region, I’ll count only occurences with as genuine.) Poor inequality. What happened?
Some takes on the issue:
- All of the other inequalities mentioned here have natural interpretations in terms of norms.
- Was the arithmetic-geometric mean inequality ever considered important in mathematics, or is it just a mirage created by maths competitions? In the former case, recur – when was it thought important and why, and when did this stop being the case? In the latter case – how did this meme enter teen maths competition culture, and why has it persisted?
- As a friend pointed out at lunchtime, the arithmetic-geometric mean inequality is equivalent to the statement that the exponential function is convex. In this guise, the inequality lives on.
Other possibilities? Should we put this to a vote?