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It is well known that, if HM denotes the harmonic mean and AM the arithmetic mean, we have $$ AM(x) \ge HM(x) $$

Now I am dealing with the expression $$ \frac{1}{HM(x)} - \frac{1}{AM(x)} $$ A trivial lower bound for this expression is $0$, but is there also a nice upper bound?

Cheers!

EDIT: Or, if there's no general upper bound, might there be one if all numbers involved are positive?

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1 Answer

up vote 2 down vote accepted

No, consider just two numbers.

$$\frac{x+y}{2xy} - \frac{2}{x+y} $$

Fix $x \gt 0$ and as $y \to 0+$, this is unbounded.

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Could it be that you've got a typo in your answer? Because if I plug $\pi/2$ into the first formula above, I don't see how this would be unbounded. –  Lagerbaer Jan 26 '11 at 22:54
    
@Lager: Yes there was typo. It is $a \to \pi/2$. –  Aryabhata Jan 26 '11 at 22:56
    
@Lager: I also had the formula for HM wrong. I have corrected. –  Aryabhata Jan 26 '11 at 23:03
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