Let *x*_{n} be a sequence of non-negative numbers. Then the sum of their running geometric means is bounded by *e* times their sum. In symbols

The inequality is strict unless all the *x*‘s are zero, and the constant *e* on the right side is optimal. Torsten Carleman proved this theorem in 1923.

Interesting. I guess you need to require convergence of the sum, though.

If the sum diverges, the right side is ∞ and so the inequality holds.

However, the statement after the equation does require convergence: “The inequality is strict unless all the

x‘s are zero, and the constant e on the right side is optimal.” If allxare 1, for example, both sides diverge and so in a sense you have equality, not a strict inequality.Interesting. It’s one of those things in math that are odd to me on several levels:

1) Why would that be the case? The relationship between the 2 values isn’t obvious at all.

2) Ok, I’ll admit, the guy proved it, so it is the case. WHY did he even think of proving it? :)

3) Finally, even though I’m sure I wouldn’t understand the answer, I’m curious as to HOW he proved it.

It’s not often one of these mathematical curiosities has all these levels of puzzlement for me :)

There’s a bit more info about it at http://mathworld.wolfram.com/CarlemansInequality.html.

The result looks almost divine at first look.

But has it any use somewhere ? Was it used to prove any other result ?

Where did you meet it in the world of math or is it a lonely formula ?