Bound on deviation between arithmetic and harmonic mean?

Question

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?

Answer 1

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.