Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?


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

share|improve this question

1 Answer 1

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.

share|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.